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Published online by Cambridge University Press:  14 December 2022

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Currently the two popular ways to practice Robinson’s nonstandard analysis are the model-theoretic approach and the axiomatic/syntactic approach. It is sometimes claimed that the internal axiomatic approach is unable to handle constructions relying on external sets. We show that internal frameworks provide successful accounts of nonstandard hulls and Loeb measures. The basic fact this work relies on is that the ultrapower of the standard universe by a standard ultrafilter is naturally isomorphic to a subuniverse of the internal universe.

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1 Introduction

Robinson named his theory “Non-standard Analysis since it involves and was, in part, inspired by the so-called Non-standard models of Arithmetic whose existence was first pointed out by T. Skolem” [Reference Robinson27, p. vii]. Currently there are two popular ways to practice Robinson’s nonstandard analysis.

The model-theoretic approach encompasses Robinson’s enlargements, obtained from the Compactness Theorem [Reference Robinson27], ultrapowers [Reference Luxemburg21], and nonstandard universes based on superstructures [Reference Chang and Keisler4, Reference Robinson and Zakon28].

The axiomatic/syntactic approach originated in [Reference Hrbacek9, Reference Nelson24]. Nelson’s $\mathrm {IST}$ is particularly well known (see, for example, [Reference Diener and Stroyan5, Reference Robert26]). The monograph of Kanovei and Reeken [Reference Kanovei and Reeken14] is a comprehensive reference for axiomatic nonstandard analysis.

In the model-theoretic approach one works with various nonstandard universes in $\mathrm {ZFC}$ ; see [Reference Chang and Keisler4, Section 4.4] for definitions and terminology associated with this framework. Let $\mathfrak {N} = ( V(X), V({}^{\ast }\! X), \ast )$ be a nonstandard universe. The collection ${}^{\ast }\!V(X)$ of internal sets in $\mathfrak {N}$ is isomorphic to some bounded ultrapowerFootnote 1 of the superstructure $V(X)$ . It is expanded to the superstructure $V({}^{\ast }\! X)$ that contains also external sets.

Axiomatic systems for the internal part of nonstandard set theory, formulated in the $\operatorname {\mathrm {st}}$ - $\in $ -language, are well suited for the development of infinitesimal analysis and much beyond. It is generally acknowledged that internal theories are easier to learn and to work with than the model-theoretic approaches. However, it is sometimes claimed as a shortcoming of the internal approach that external sets are essential for some of the most important new contributions of Robinsonian nonstandard analysis to mathematics, such as the constructions of nonstandard hulls [Reference Luxemburg22] and Loeb measures [Reference Loeb17]; see for example [Reference Loeb and Wolff19, p. xiv], [Reference Loeb and Wolff20, p. vii], and [Reference Loeb18, Reference Zlatoš34]. Such claims are not meant to be taken literally. As $\mathrm {IST}$ includes $\mathrm {ZFC}$ among its axioms, nonstandard universes and the external constructions in them make sense in $\mathrm {IST}$ . However, nonstandard universes have their own notions of standard and internal that are not immediately related to the analogous notions provided by $\mathrm {IST}$ . Such a duplication of concepts could be confusing and complicates attempts to work directly in superstructure frameworks inside $\mathrm {IST}$ . In any case, the issue in question is whether these constructions can be carried out in internal set theories using the notions that these theories axiomatize.

In this paper we describe an approach to external constructions that is better suited to the conceptual framework provided by internal set theories. The techniques we employ have been known for a long time, but their use for the purpose of implementing external methods in internal set theories does not seem to explicitly appear in the literature. We show that practically all objects whose construction in a superstructure involves external sets can be obtained in the internal axiomatic setting by these techniques.

We work in Bounded Set Theory $\mathrm {BST}$ (see [Reference Kanovei and Reeken14, Chapter 3]). The axioms of $\mathrm {BST}$ are a slight modification of the more familiar axioms of $\mathrm {IST}$ . We state them in Section 2 and follow with a discussion of how definable external sets can be handled in $\mathrm {BST}$ as abbreviations.

The not-so-well-known fact about $\mathrm {BST}$ is that its universe contains subuniverses isomorphic to any standard ultrapower (or even limit ultrapower) of the standard universe. For every internal set w there is a subuniverse $\mathbf {S}_w$ and a simply defined isomorphism of $\mathbf {S}_w$ with the ultrapower of the standard universe by a standard ultrafilter generated by w. Thus $\mathrm {BST}$ naturally provides an analog of the internal universe ${}^{\ast } \!V(X)$ of any superstructure. The subuniverses $\mathbf {S}_w$ satisfy some of the axioms of $\mathrm {BST}$ , as described in Section 3.1; hence one can work with these subuniverses axiomatically, without any reference to their relationship to ultrafilters or ultrapowers.

External subsets of $\mathbf {S}_w$ and higher-order external sets built from them are not objects of $\mathrm {BST}$ , but the above-mentioned isomorphism, in combination with the powerful principle of Standardization, provides a natural way to code these “non-existent” external sets by standard sets and to imitate the superstructure $V({}^{\ast }\! X)$ of external sets. This idea is developed in Section 3.2.

The heart of the paper is Section 4, where we show how to construct nonstandard hulls of standard metric and uniform spaces and Loeb measures. We also consider analogous constructions on internal normed spaces, and outline how one can treat neutrices and external numbers. From the practical point of view, the best way to use these techniques may be to work out the external constructions informally, and then code them up by standard sets. In principle, any construction involving external sets can be carried out in the framework of $\mathrm {BST}$ by these methods.

Section 5 develops the relationship between the subuniverses $\mathbf {S}_w$ and ultrapowers, and supplies proofs of the properties of $\mathbf {S}_w$ stated in Section 3.1.

In Section 6 we review some ways that have been proposed for doing external constructions in $\mathrm {IST}$ previously and discuss their shortcomings. The principal difficulty is that they tend to produce objects that are “too large” to be suitable for further work (larger than the class of standard elements of any standard set).

2 The internal set theory $\mathrm {BST}$

We work in the framework of $\mathrm {BST}$ .Footnote 2 The theory $\mathrm {BST}$ is formulated in the $\operatorname {\mathrm {st}}$ - $\in $ -language. It is a conservative extension of $\mathrm {ZFC}$ .

Quantifiers with the superscript $\operatorname {\mathrm {st}}$ range over standard sets. Quantifiers with the superscript $\operatorname {\mathrm {st}}\! \operatorname {\mathrm {fin}}$ range over standard finite sets. The axioms of $\mathrm {BST}$ are, in addition to $\mathrm {ZFC}$ (the $\mathrm {ZFC}$ axiom schemata of Separation and Replacement apply to $\in $ -formulas only):

  • $\mathrm {B}$ (Boundedness) $\quad \forall x \, \exists ^{\operatorname {\mathrm {st}}} y \,(x \in y)$ .

  • $\mathrm {T}$ (Transfer) Let $\phi (v)$ be an $\in $ -formula with standard parameters. Then

    $$ \begin{align*} \forall^{\operatorname{\mathrm{st}}} x\; \phi(x) \rightarrow \forall x\; \phi(x). \end{align*} $$
  • $\mathrm {S}$ (Standardization) Let $\phi (v)$ be an $\operatorname {\mathrm {st}}$ - $\in $ -formula with arbitrary parameters. Then

    $$ \begin{align*} \forall A\, \exists^{\operatorname{\mathrm{st}}} S\; \forall^{\operatorname{\mathrm{st}}} x\; (x \in S \longleftrightarrow x \in A \,\wedge\,\phi(x)). \end{align*} $$
  • $\mathrm {BI}$ (Bounded Idealization) Let $\phi (u,v)$ be an $\in $ -formula with arbitrary parameters. For every standard set A

    $$\begin{align*}\forall^{\operatorname{\mathrm{st}}\!\operatorname{\mathrm{fin}}} a \subseteq A \;\exists y\; \forall x \in a\; \phi(x,y) \longleftrightarrow \exists y \; \forall^{\operatorname{\mathrm{st}}} x \in A \; \phi(x,y). \end{align*}$$

See the references [Reference Fletcher, Hrbacek, Kanovei, Katz, Lobry and Sanders7, Reference Kanovei and Reeken14] for motivation and more detail.

An equivalent existential version of Transfer is

$$ \begin{align*} \exists x\; \phi(x) \rightarrow \exists^{\operatorname{\mathrm{st}}} x\; \phi(x), \end{align*} $$

for $\in $ -formulas $\phi $ with standard parameters. Yet another equivalent version, easily obtained by induction on the logical complexity of the $\in $ -formula $\phi (v_1,\ldots ,v_k)$ (with standard parameters), is

$$ \begin{align*} \forall^{\operatorname{\mathrm{st}}} x_1,\ldots, x_k\; [\,\phi (x_1,\ldots, x_k) \longleftrightarrow \phi^{\operatorname{\mathrm{st}}} (x_1,\ldots, x_k)], \end{align*} $$

where $\phi ^{\operatorname {\mathrm {st}}}$ is the formula obtained from $\phi $ by relativizing all quantifiers to $\operatorname {\mathrm {st}}$ (that is, by replacing each occurrence of $\exists $ by $\exists ^{\operatorname {\mathrm {st}}} $ and each occurrence of $\forall $ by $\forall ^{\operatorname {\mathrm {st}} } $ ).

As usual in mathematics, symbols $\mathbb {N}$ and $\mathbb {R}$ denote respectively the set of all natural numbers and the set of all reals. In internal set theories there are two ways of thinking about them. In the “internal picture” $\mathbb {R}$ is viewed as the usual set of reals in which the predicate $\operatorname {\mathrm {st}}$ singles out some elements as “standard”; similarly for any infinite standard set. This is the view familiar from [Reference Nelson24]. In the “standard picture” the usual set $\mathbb {R}$ is viewed as containing, in addition to its standard elements, also fictitious, ideal elements. See [Reference Fletcher, Hrbacek, Kanovei, Katz, Lobry and Sanders7] for further discussion.

Mathematics in $\mathrm {BST}$ can be developed in the same way as in $\mathrm {IST}$ . In particular, real numbers $r, s$ are infinitely close (notation: $r \simeq s$ ) if $|r - s| < 1/n$ holds for all standard $n \in \mathbb {N}\setminus \{0\}$ . A real number $r $ is an infinitesimal if $r \simeq 0$ , $r \ne 0$ . It is limited if $|r| < n$ for some standard $n \in \mathbb {N}$ . We recall that for every limited $x \in \mathbb {R}$ there is a unique standard $r \in \mathbb {R}$ such that $r \simeq x$ ; it is called the shadow (or standard part) of x and denoted $\operatorname {\mathrm {sh}}(x)$ ; we also define $\operatorname {\mathrm {sh}}(x) = +\infty $ when x is unlimited, $x> 0$ , and $\operatorname {\mathrm {sh}}(x) = -\infty $ when x is unlimited, $x < 0$ .

All objects whose existence is postulated by $\mathrm {BST}$ are sets, sometimes called internal sets for emphasis. The Separation axiom holds for $\in $ -formulas only. But it is common practice in the literature based on the internal axiomatic approach to introduce definable external sets as convenient abbreviations (see, e.g., [Reference Diener and Stroyan5, Reference Vakil31]). One can enrich the language of the theory by names for extensions of arbitrary $\operatorname {\mathrm {st}}$ - $\in $ -formulas and in this way talk about $\operatorname {\mathrm {st}}$ - $\in $ -definable subclasses of the universe of all (internal) sets. We note that (a) this does not amount to a formalization of a new type of entity called “external set,” which is a more complicated task; and (b) this does not amount to informal use of the term “external set,” either (in the sense of relying on a background formalization). It is similar to the way set theorists routinely employ classes in $\mathrm {ZFC}$ (the class of all sets, the class of all ordinals). Such classes serve as convenient shortcuts in mathematical discourse because one can work with them “as if” they were objects, but they can in principle be replaced by their defining formulas. Example 2.1 illustrates this familiar procedure.

Let $\phi (v)$ be an $\operatorname {\mathrm {st}}$ - $\in $ -formula with arbitrary parameters. We employ dashed curly braces to denote the class . We emphasize that this is merely a matter of convenience; the expression is just another notation for $\phi (z)$ . Usually we denote classes by boldface characters. Those classes that are included in some set are external sets. If there is a set A such that $\forall x\,(x \in A \longleftrightarrow \phi (x))$ , then the class can be identified with the set A.Footnote 3 Monads and galaxies are some familiar examples of external sets that are usually not sets. Let $(M, d) $ be a metric space: the monad of $a \in M$ is , and the galaxy of $a \in M$ is . Some useful proper classes (i.e., classes that are not external sets) are (the universe of all (internal) sets),Footnote 4 and (the universe of all standard sets).

Example 2.1. Let M, $f: M \to M$ and $a \in M$ be standard. A convenient way of defining continuity is as follows: The function f is continuous at a if $f[\mathbf {M}(a) ]\subseteq \mathbf {M}(f(a)).$ But the use of external sets in this definition can be eliminated by rephrasing it as: The function f is continuous at a if for all x, $ d(x,a) \simeq 0$ implies $d(f(x),f(a)) \simeq 0$ .

Definable external collections of internal sets are adequately handled in $\mathrm {BST}$ in this manner; we refer to [Reference Vakil31] for a thorough discussion. Difficulties arise only when higher-level constructs on external sets are needed, such as quotient spaces and power sets. We show how to handle such difficulties in Sections 3.2, 4.1, 4.4, and 4.5.

3 Subuniverses of the universe of $\mathrm {BST}$

3.1 w-standard sets

Let us fix a set w and a standard set I such that $w \in I$ .Footnote 5

Definition 3.1. A set x is called w-standard (notation: $\operatorname {\mathrm {st}}_w(x)$ ) if $x = f(w)$ for some standard function f with domain I. We let be the universe of all w-standard sets. The notion of w-standardness depends only on w, not on the choice of the standard set I.

Proposition 3.2.

  1. (a) $\quad \forall x\, (\operatorname {\mathrm {st}}(x) \rightarrow \operatorname {\mathrm {st}}_w(x))$ .

  2. (b) $\forall x\, (\operatorname {\mathrm {st}}_w(x) \rightarrow \operatorname {\mathrm {st}}(x))$ holds if and only if w is standard.

  3. (c) $\operatorname {\mathrm {st}}_w(f) \,\wedge \, \operatorname {\mathrm {st}}_w(x) \,\wedge \, x \in \operatorname {dom} f\rightarrow \operatorname {\mathrm {st}}_w(f(x))$ .


  1. (a) Let x be standard; we have $x = c_x(w)$ where $c_x$ is the constant function with value x.

  2. (b) If w is standard, then every $f(w)$ for standard f is standard. If w is nonstandard, let $f(i) =i$ be the identity function on I. Then $f(w) = w$ is w-standard but not standard.

  3. (c) Assume $\operatorname {\mathrm {st}}_w (f)$ , $\operatorname {\mathrm {st}}_w(x)$ , and $x \in \operatorname {dom} f$ . Then there are standard functions F, G on I such that $f = F(w)$ and $x = G(w)$ . Define a function H on I by

    $$ \begin{align*}\kern-7pt H(i) = F(i) (G(i)) \text{ when the right side is defined; } H(i) = \emptyset \text{ otherwise.} \end{align*} $$

Then H is a standard function on I and $H(w) = f(x)$ .

Definition 3.3. A set w is good if there is $\nu \in \mathbb {N}$ such that $\nu $ is w-standard but not standard.

In particular, if $\nu \in \mathbb {N}$ is nonstandard, then $w = \nu $ is good and, more generally, $w = \langle \nu , z \rangle $ is good for any set z.

The following facts are immediate consequences of known results (see [Reference Kanovei and Reeken14, Sections 3.3, 6.1, and 6.2, esp. Theorem 6.2.6]). For easy reference we give the proofs in Section 5. Quantifiers with the superscript $\operatorname {\mathrm {st}}_w$ range over w-standard sets.

Proposition 3.4.

  1. (1) (Transfer from w-standard sets) Let $\phi $ be an $\in $ -formula with w-standard parameters. Then

    $$ \begin{align*}\forall^{\operatorname{\mathrm{st}}_w} x\; \phi(x) \rightarrow \forall x\; \phi(x).\end{align*} $$
  2. (2) (Countable Idealization into w-standard sets)

    Let $\phi $ be an $\in $ -formula with w-standard parameters. If w is good, then

    $$ \begin{align*} \kern-9pt\forall^{\operatorname{\mathrm{st}}} n \in \mathbb{N}\;\exists x\; \forall m \in \mathbb{N}\; (m \le n \;\rightarrow \phi(m,x))\longleftrightarrow \exists^{\operatorname{\mathrm{st}}_w} x \; \forall^{\operatorname{\mathrm{st}}} n \in \mathbb{N} \; \phi(n,x).\end{align*} $$
    In other words, $(\mathbf {S}_w, \mathbf {S}, \in )$ satisfies Countable Idealization.
  3. (3) (Representability)

    Let $\phi (v)$ be an $\in $ -formula with standard parameters. If $I $ is standard, $w \in I$ , x is w-standard, and $\phi (x)$ holds, then there is a standard function f with $\operatorname {dom} f =I$ such that $x = f(w)$ and $\phi (f(i))$ holds for all $i \in I$ .

An equivalent formulation of (1) is Transfer into w-standard sets:

$$ \begin{align*}\exists x\; \phi(x) \rightarrow \exists^{\operatorname{\mathrm{st}}_w} x\; \phi(x).\end{align*} $$

Another equivalent formulation for $\phi (v_1,\ldots ,v_r)$ with w-standard parameters is

$$ \begin{align*} \forall^{\operatorname{\mathrm{st}}_w} x_1,\ldots, x_r\; [\,\phi (x_1,\ldots, x_r)\longleftrightarrow \phi^{\operatorname{\mathrm{st}}_w} (x_1,\ldots, x_r)], \end{align*} $$

where $\phi ^{\operatorname {\mathrm {st}}_w}$ is the formula obtained from $\phi $ by relativizing all quantifiers to $\operatorname {\mathrm {st}}_w$ . In yet other words, $(\mathbf {V}, \mathbf {S}_w, \in )$ satisfies Transfer. It also satisfies Boundedness and, for good w, Bounded Idealization (see [Reference Kanovei and Reeken14, Theorem 6.1.16(i)]), but not Standardization (ibid, Theorem 6.1.15). Although we do not need these results in this paper, together they show that $(\mathbf {V}, \mathbf {S}_w, \in )$ satisfies all the axioms of $\mathrm {BST}$ except Standardization. Note that here $\mathbf {S}_w$ plays the role of a new “thick standard universe.” On the other hand, $(\mathbf {S}_w , \mathbf {S} , \in )$ satisfies Transfer, Boundedness, Standardization, and Countable Idealization; here $\mathbf {S}_w$ plays the role of a new “thin internal universe.”

Idealization can be strengthened from $\mathbb {N}$ to sets of cardinality $\kappa $ if w is chosen carefully. We leave the technical definition of $\kappa ^+$ -good sets to Section 5.2 (see Definition 5.7). For our applications we need only to know that for every standard uncountable cardinal $\kappa $ and every z there exist $\kappa ^+$ -good w so that z is w-standard, a result which is also proved there.

Proposition 3.5 (Idealization into w-standard sets over sets of cardinality  $\le \kappa $ ).

Let $\phi $ be an $\in $ -formula with w-standard parameters. If w is $\kappa ^+$ -good, then for every standard set A of cardinality $\le \kappa $ ,

$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}\! \operatorname{\mathrm{fin}}} a \subseteq A \;\exists y\; \forall x \in a\; \phi(x,y) \longleftrightarrow \exists^{\operatorname{\mathrm{st}}_w} y \; \forall^{\operatorname{\mathrm{st}}} x \in A \; \phi(x,y).\end{align*} $$

Propositions 3.4 and 3.5 do not exhaust the properties of the universes $\mathbf {S}_w$ ; for a list of further useful principles see [Reference Kanovei and Reeken14, Theorem 3.3.7].

3.2 Coding external sets

Definition 3.1 provides a natural way to represent w-standard sets by standard functions: A w-standard $\xi \in \mathbf {S}_w$ is represented by any standard $f \in \mathbf {V}^I$ such that $f(w) = \xi $ . Note that every $\xi $ has a proper class of representations. This causes some technical difficulties (see Section 5), which for our purposes are best resolved by fixing a universal standard set V so that all objects of interest are subsets of V or relations on V; usually one requires $\mathbb {R}\subseteq V$ . By Representability, for every $\xi \in V \cap \mathbf {S}_w$ there exists a standard $f \in V^I$ such that $f(w) = \xi $ . Moreover, if $\psi $ is an $\in $ -formula with standard parameters and $\psi (\xi )$ holds, then f can be chosen so that $\psi (f(i))$ holds for all $i \in I$ . The representation makes possible a coding of the external subsets of $V \cap \mathbf {S}_w$ by standard sets, and the coding process can be continued to the putative higher levels of the external cumulative hierarchy. This process is enabled by the principle of Standardization.

Definition 3.6. Let $\phi (v)$ be a formula in the $\operatorname {\mathrm {st}}$ - $\in $ -language, with arbitrary parameters. We let

$$ \begin{align*}{}^{\operatorname{\mathrm{st}}} \{ x \in A \,\mid\, \phi (x) \}\end{align*} $$

denote the standard set S such that $\forall ^{\operatorname {\mathrm {st}}} x\,(x \in S \longleftrightarrow x \in A \,\wedge \,\phi (x))$ .

The principle of Standardization postulates the existence of this set and Transfer guarantees its uniqueness. Also by Transfer, if $\psi $ is any $\in $ -formula with standard parameters and $\forall ^{\operatorname {\mathrm {st}}} x \in A\; (\phi (x) \rightarrow \psi (x))$ , then $\forall x \in S\; \psi (x)$ .

In particular, if $\phi (u.v)$ is a formula in the $\operatorname {\mathrm {st}}$ - $\in $ -language with arbitrary parameters, A, B are standard, and for every standard $x \in A$ there is a unique standard $y \in B$ such that $\phi (x, y)$ , then there is a unique standard function $F: A \to B$ such that $\forall ^{\operatorname {\mathrm {st}}} x \in A\;\phi (x, F(x))$ holds. It suffices to let $F = {}^{\operatorname {\mathrm {st}}}\{ \langle x, y \rangle \in A \times B \,\mid \, \phi (x,y) \}$ .

Definition 3.7. The $w,V$ -code of an external set $\mathbf {X}\subseteq V \cap \mathbf {S}_w$ is the standard set

$$ \begin{align*}\Psi_{w,V} (\mathbf{X}) =X = {}^{\operatorname{\mathrm{st}}}\{ f \in V^I\,\mid\, f(w) \in \mathbf{X} \}.\end{align*} $$

In words, $\Psi _{w,V} (\mathbf {X}) $ is the standard set whose standard elements are precisely the standard $f\in V^I $ with $f(w) \in \mathbf {X}$ .

We omit the subscripts w and/or V when they are understood from the context.

The coding is trivially seen to preserve elementary set-theoretic operations.

Proposition 3.8. For any external sets $\mathbf {X}_1, \mathbf {X}_2 \subseteq V \cap \mathbf {S}_w$ :

  1. (1) $\Psi (\emptyset ) = \emptyset $ , $\mathbf {X}_1 \subseteq \mathbf {X}_2 \longleftrightarrow \Psi (\mathbf {X}_1) \subseteq \Psi (\mathbf {X}_2) $ ;

  2. (2) $\Psi (\mathbf {X}_1 \cup \mathbf {X}_2) =\Psi (\mathbf {X}_1) \cup \Psi (\mathbf {X}_2) $ , $\Psi (\mathbf {X}_1 \cap \mathbf {X}_2) = \Psi (\mathbf {X}_1) \cap \Psi (\mathbf {X}_2) $ ;

  3. (3) $\Psi (\mathbf {X}_1 \setminus \mathbf {X}_2) = \Psi (\mathbf {X}_1) \setminus \Psi (\mathbf {X}_2) .$

The coding preserves infinite unions and intersections as well. An externally countable sequence of external subsets of V can be viewed in $\mathrm {BST}$ as an external subset $\mathbf {X}$ of $\mathbb {N} \times V$ , with for standard $n \in \mathbb {N}$ . Let $\langle X_n \,\mid \, n \in \mathbb {N} \rangle $ be the standard sequence such that $X_n = \Psi (\mathbf {X}_n)$ holds for all standard n (its existence follows from Standardization). Then

$$ \begin{align*} \Psi(\bigcup_{n \in \mathbb{N} \cap \mathbf{S}} \mathbf{X}_n) = \bigcup_{n \in \mathbb{N}} X_n \end{align*} $$

because if $f \in \bigcup _{n \in \mathbb {N}} X_n$ is standard, then the least $n \in \mathbb {N}$ such that $f \in X_n$ is standard. Similarly,

$$ \begin{align*}\Psi(\bigcap_{n \in \mathbb{N} \cap \mathbf{S}} \mathbf{X}_n) = \bigcap_{n \in \mathbb{N}} X_n \end{align*} $$

because if a standard $f \in \Psi (\bigcap _{n \in \mathbb {N} \cap \mathbf {S}} \mathbf {X}_n)$ , then $f \in X_n$ for all standard $n \in \mathbb {N}$ , and hence $f \in \bigcap _{n \in \mathbb {N}} X_n$ by Transfer.

It is convenient to relax the definition of coding so that every standard $S \subseteq V^I$ is a code of some external $\mathbf {X} \subseteq V \cap \mathbf {S}_w$ .

Definition 3.9. A standard set S codes $\mathbf {X} \subseteq V \cap \mathbf {S}_w$ if $f(w) \in \mathbf {X}$ for each standard $f \in S$ and for each $\xi \in \mathbf {X}$ there is some standard $f \in S$ such that $f(w) = \xi $ .

If S codes $\mathbf {X}$ , then $\Psi (\mathbf {X}) = {}^{\operatorname {\mathrm {st}}} \{f \in V^I \,\mid \, f(w) = g(w) \text { for some standard} g \in S\}.$ A code of $\mathbf {X}$ has to contain a representation for each $\xi \in \mathbf {X}$ , but not necessarily all such representations.

We note that coding is independent of the choice of the universal standard set in the following sense: If $\mathbf {X} \subseteq V_1 \cap \mathbf {S}_w$ and $V_1 \subseteq V_2$ , then S codes $\mathbf {X}$ viewed as a subset of $V_1 \cap \mathbf {S}_w$ iff S codes $\mathbf {X}$ viewed as a subset of $V_2 \cap \mathbf {S}_w$ . Hence the exact choice of V is usually of little importance.

For any set x, let $c_x$ be the constant function with value x. The informal identification of x with $c_x$ enables the identification of a standard set A with a code $ {}^{\operatorname {\mathrm {st}}}\{ c_x \,\mid \, x \in A \} = \{ c_x \,\mid \, x \in A \}$ for $A \cap \,\mathbf {S}$ . On the other hand, the standard set $A^I$ is a code for $A \cap \,\mathbf {S}_w$ .

Every standard $S \subseteq V^I$ is a code of the unique external set

Such S codes a subset of an external set $\mathbf {X} $ iff $\forall ^{\operatorname {\mathrm {st}}} f\, (f \in S \rightarrow f(w) \in \mathbf {X})$ iff $S \subseteq \Psi (\mathbf {X}) $ . Consequently, it would make sense to interpret the power set of $\widetilde {X}=\Psi (\mathbf {X}) $ as a code for the “non-existent” external power set $\mathcal {P}^{\operatorname {\mathrm {ext}}} (\mathbf {X})$ of $\mathbf {X}$ , and the standard subsets of $\mathcal {P}(\widetilde {X})$ as codes for the “external subsets of $\mathcal {P}^{\operatorname {\mathrm {ext}}} (\mathbf {X})$ .” Since $\mathrm {BST}$ does not allow collections of external sets, this last remark cannot be made rigorous in it, but one can proceed “as if” such higher-order external sets existed.

Intuitively, there is a hierarchy of external sets built up over $V \cap \mathbf {S}_w$ :

$$ \begin{align*} \boldsymbol{\mathcal{H}}_1 = \mathcal{P}^{\operatorname{\mathrm{ext}}}(V \cap \mathbf{S}_w) \text{ and } \boldsymbol{\mathcal{H}}_{n+1} = \mathcal{P}^{\operatorname{\mathrm{ext}}} (\boldsymbol{\mathcal{H}}_n) \text{ for standard } n \in \mathbb {N}\end{align*} $$

(we stop here to avoid further complications). This hierarchy cannot be formalized in $\mathrm {BST}$ . But the corresponding hierarchy over $V^I$ is well-defined:

$$ \begin{align*} H_1 = \mathcal{P}(V^I) \text{ and } H_{n+1} = \mathcal{P} (H_n) \text{ for standard } n \in \mathbb{N}. \end{align*} $$

In $\mathrm {BST}$ one can work legitimately in the latter hierarchy while keeping in mind that the coding establishes a (many-one) correspondence between $\boldsymbol {\mathcal {H}}_n$ and $H_n \cap \mathbf {S}$ . Both hierarchies and their relationship could be formalized in $\mathrm {HST}$ , a conservative extension of $\mathrm {BST}$ to a theory that encompasses also external sets [Reference Hrbacek9, Reference Kanovei and Reeken14].

Note that the coding process treats elements $\xi $ of $V \cap \mathbf {S}_w$ as individuals; they are not coded by $\Psi $ . Thus $\xi $ is represented by any f with $f(w) = \xi $ , but $\Psi (\xi )$ is undefined. The set $\{\xi \} \subseteq V\cap \mathbf {S}_w$ is coded by $\{f\}$ , even when $ \{\xi \} \in V\cap \mathbf {S}_w$ . In particular, if $\mathbf {R}$ is a binary relation on $V \cap \mathbf {S}_w$ , then the code for $\mathbf {R}$ is the standard binary relation

$$ \begin{align*}\Psi (\mathbf{R}) =R = \,{}^{\operatorname{\mathrm{st}}}\{ \langle f , g \rangle \,\mid\, f, g \in V^I \,\wedge\, \langle f(w), g(w)\rangle \in \mathbf{X} \}.\end{align*} $$

Similarly for functions and relations of higher arity.

Another variant of coding represents each $\xi \in \mathbf {X}$ by a single object.

Definition 3.10. Let $f \in V^I$ be standard. We let

$$ \begin{align*}f_{w,V} = f_w ={}^{\operatorname{\mathrm{st}}}\{g \in V^I\,\mid\, g(w) = f(w) \}.\end{align*} $$

We define standard sets $V^I/w = {}^{\operatorname {\mathrm {st}}}\{ f_w \,\mid \, f \in V^I\}$ and, for $\mathbf {X} \subseteq V \cap \mathbf {S}_w,$

$$ \begin{align*} \widetilde{\Psi}_w (\mathbf{X}) &= {}^{\operatorname{\mathrm{st}}} \{ f_w \in V^I/w \,\mid \, f(w) \in \mathbf{X}\} \\ &= {}^{\operatorname{\mathrm{st}}} \{ F \in V^I/w \,\mid\, \exists^{\operatorname{\mathrm{st}}} f \in V^I\, (F = f_w \,\wedge\, f(w) \in \mathbf{X}) \}. \end{align*} $$

The coding $\widetilde {\Psi }_w$ is one–one. For standard $X \subseteq V^I/w $ we let

One can obtain $\widetilde {\Psi }_w (\mathbf {X}) $ from $\Psi _w (\mathbf {X}) $ and vice versa:

$$ \begin{align*} \widetilde{\Psi}_w (\mathbf{X}) = {}^{\operatorname{\mathrm{st}}} \{ f_w \,\mid\, f \in \Psi_w(\mathbf{X})\} \text{ and } \Psi_w(\mathbf{X}) = {}^{\operatorname{\mathrm{st}}} \{ f \in V^I \,\mid\, f_w \in \widetilde{\Psi}_w (\mathbf{X}) \}. \end{align*} $$

Proposition 3.8 and the subsequent paragraph hold with $\Psi $ replaced by $\widetilde {\Psi }$ .

We define the hierarchy $\widetilde {H}_1 = \mathcal {P}(V^I/w)$ and $\widetilde {H}_{n+1} = \mathcal {P} (\widetilde {H}_n)$ for standard $n \in \mathbb {N}$ . The coding $\widetilde {\Psi }_w$ maps $\boldsymbol {\mathcal {H}}_1$ onto $\widetilde {H}_1 \cap \mathbf {S}$ . It extends informally to higher levels by $\widetilde {\Psi }_w(\mathbf {X}) = {}^{\operatorname {\mathrm {st}}} \{ \widetilde {\Psi }_w(\mathbf {Y}) \,\mid \, \mathbf {Y} \in \mathbf {X}\}$ and provides a one–one correspondence between $\boldsymbol {\mathcal {H}}_n$ and $\widetilde {H}_n \cap \mathbf {S}$ .

4 Nonstandard hulls and Loeb measures in $\mathrm {BST}$

4.1 Nonstandard hulls of standard metric spaces

Let $\mathbb {R}$ be the field of real numbers and let $(M, d)$ be a standard metric space, so that M is a standard set and the distance function is a standard mapping $d: M \times M \to \mathbb {R}$ . A point $x \in M$ is finite if $d(x,p)$ is limited for some (equivalently, for all) standard $p \in M$ .

Fix an unlimited integer $w \in \mathbb {N}$ . We define the standard set $B_w$ by

$$ \begin{align*}B_w =\, {}^{\operatorname{\mathrm{st}}} \{ f \in M^{\mathbb{N}} \,\mid\, f(w) \text{ is a finite point of }M\}.\end{align*} $$

The standard relation $E_w$ on $B_w$ is defined by

$$ \begin{align*}E_w = \, {}^{\operatorname{\mathrm{st}}} \{ \langle f, g \rangle \in B _w\times B_w \,\mid\, d(f(w) , g(w))\simeq 0\}.\end{align*} $$

Clearly $E_w $ is reflexive, symmetric, and transitive on standard elements of $B_w$ ; it follows by Transfer that $E_w$ is an equivalence relation on $B_w$ . We denote the equivalence class of f modulo $E_w$ by $f_{E_w}$ .

By Standardization, there is a standard function $D_w: B_w \times B_w \to \mathbb {R}$ determined by the requirement that $D_w ( f, g ) = \operatorname {\mathrm {sh}} ( d (f(w) , g(w) ))$ for all standard $f, g\in B_w$ .

For standard $f, g\in B_w$ the distance

$$ \begin{align*}d (f(w) , g(w) )) \le d(f(w), f(0)) + d(f(0), g(0)) + d(g(0), g(w))\end{align*} $$

is limited, so $\operatorname {\mathrm {sh}} ( d (f(w) , g(w) )) \in \mathbb {R}$ .

For standard $f,g,h \in B_w$ clearly $D_w ( f, g ) = D_w ( g,f ) $ ,

$$ \begin{align*}D_w ( f, h ) \le D_w ( f, g ) + D_w ( g, h) , \text{ and } D_w ( f, g ) = 0 \text{ iff } \langle f, g \rangle \in E_w.\end{align*} $$

Also, for standard $f,g,f',g' \in B_w$ we have

$$ \begin{align*}\bigl( \langle f, f'\rangle \in E_w \,\wedge \, \langle g, g' \rangle \in E_w \bigr)\rightarrow D_w(f,g) = D _w(f', g'). \end{align*} $$

By Transfer these properties hold for all $f,g ,h, f',g' \in B_w$ .

We observe that, for the natural choice $V = M \cup \mathbb {R}$ , $B_w$ is a code for the external set

$E_w$ is nothing but a code for the external equivalence relation

and $D_w$ is a code for the external function $\mathbf {D}_w: \mathbf {B}_w \times \mathbf {B}_w \to \mathbb {R} \cap \mathbf {S}$ defined by


The construction of the nonstandard hull of $(M,d)$ by the usual external method would form the quotient space $\mathbf {B} _w/\mathbf {E}_w$ of $\mathbf {B}_w$ modulo $\mathbf {E}_w$ . This step cannot be carried out in $\mathrm {BST}$ . For $x \in \mathbf {B}_w$ , the equivalence class of x modulo $\mathbf {E}_w $ is , but the collection of the classes $\mathbf {M}_w(x)$ for all $x \in \mathbf {B}_w$ is not supported by $\mathrm {BST}$ . However, for standard $f \in B_w$ , the set $f_{E_w}$ is a code of $\mathbf {M}_w(x)$ when $f(w) = x$ . Therefore $\mathbf {B}_w /\mathbf {E}_w$ can be replaced by $B_w/E_w$ , which is a standard set in $\mathrm {BST}$ . The standard elements of $B_w/E_w$ are precisely the codes of the monads $\mathbf {M}_w(x)$ for $x \in \mathbf {B}_w$ . Hence $B_w/E_w$ is a code (as discussed in Section 3.2) of the “non-existent” (in $\mathrm {BST}$ ) quotient space $\mathbf {B} _w/\mathbf {E}_w$ .

We stress that while external sets serve as a motivation for our constructions, they are not actually used in them; the constructions deal only with sets of $\mathrm {BST}$ . The same applies to the rest of this section.

We let $\widehat {M}_w = B_w / E_w $ be the standard quotient space of $B_w$ modulo $E_w$ . From now on we often omit the subscript w when it is understood from the context.

The function $D $ factors by E to a (standard) function $\widehat {D} = D / E$ on $\widehat {M}$ , defined by $\widehat {D} ( f_E, g_E ) = D(f, g)$ (as shown above, the value of $\widehat {D}$ is independent of the choice of representatives f, g). It is clear that $\widehat {D}$ is a metric on $\widehat {M} $ .

The embedding c of M into $\widehat {M}$ is via constant functions: for $x \in M$ , $c(x) = (c_x)_E$ where $c_x$ is the constant function on $\mathbb {N}$ with value x. Trivially, the embedding c preserves the metrics. We identify M with its image in $\widehat {M}$ under this embedding.

We emphasize that the structure $(\widehat {M} , \widehat {D} \,)$ depends on the choice of the parameter w, an unlimited integer. A metric space has a unique completion up to isometry, but it may have many non-isometric nonstandard hulls.

Example 4.1. Let $M = \mathbb {Q}$ be the set of rationals and $d(x,y) = |x-y|$ be the usual metric on $\mathbb {Q}$ . We fix an unlimited $w\in \mathbb {N}$ . We note that, for standard f,

$$ \begin{align*}f \in B_w \longleftrightarrow f \in \mathbb{Q}^{\mathbb{N}} \,\wedge\, f(w) \text{ is limited},\end{align*} $$

and for standard $f,g \in B_w$

$$ \begin{align*}\langle f, g \rangle &\in E_w \longleftrightarrow f(w) \simeq g(w) \longleftrightarrow f(w),\\ g(w) &\in \mathbf{M}(a) \text{ for } a = \operatorname{\mathrm{sh}}(f(w)) = \operatorname{\mathrm{sh}}(g(w)).\end{align*} $$

By Standardization, there is a standard function $\Gamma : B_w/E_w \to \mathbb {R}$ such that, for standard f, $\Gamma (f/E ) = \operatorname {\mathrm {sh}}(f(w)).$ It is easy to verify that $\Gamma \upharpoonright (B_w/E_w) \cap \mathbf {S}$ is a one–one mapping of $\widehat {M}_w \cap \mathbf {S}$ onto $\mathbb {R} \cap \mathbf {S}$ that preserves the metrics:

$$ \begin{align*}\widehat{D}(f/E, g/E) &= D(f,g) = \operatorname{\mathrm{sh}}(|f(w) - g(w)|)\\ & = |\operatorname{\mathrm{sh}}(f(w)) - \operatorname{\mathrm{sh}}(g(w))| = |\Gamma(f) - \Gamma(g)|.\end{align*} $$

Moreover, $\Gamma ((c_x)_E) = x$ for any standard $x \in \mathbb {Q}$ . By Transfer, $\Gamma $ is an isometric isomorphism of $\widehat {M}_w$ and $\mathbb {R}$ which is the identity on $\mathbb {Q}$ . In particular, the nonstandard hull is independent of the choice of w.

Example 4.2. Let $M= \mathbb {N}$ and let d be the discrete metric on M; i.e., $d(x,z) = 1$ for all $x,z \in M$ , $x \neq z$ . As all points of M are finite with respect to this metric, we have $B_w = \mathbb {N}^{\mathbb {N}}$ . Also

$$ \begin{align*}E_w = {}^{\operatorname{\mathrm{st}}}\{ \langle f, g \rangle \in \mathbb{N}^{\mathbb{N}} \times \mathbb{N}^{\mathbb{N}} \,\mid\, f(w) = g(w)\}\end{align*} $$

and, for standard $f \in \mathbb {N} ^{\mathbb {N}}$ , $f/E = {}^{\operatorname {\mathrm {st}}} \{ g \in \mathbb {N}^{\mathbb {N}} \,\mid \, g(w) = f(w)\} = f_w$ . The space M is identified with a subset of $\widehat {M}_w$ via $\Gamma : (c_x)_w \mapsto x$ .

In Remark 5.5 we establish that $\widehat {M}_w =B_w/E_w = \mathbb {N}^{\mathbb {N}}/w$ is exactly the ultrapower of M by $U_w$ , an ultrafilter over $\mathbb {N}$ generated by w; in particular, it has the cardinality of the continuum. Also, $\widehat {D}$ is the discrete metric on $\widehat {M}_w$ . By replacing $\mathbb {N}$ with I as in Remark 4.6 one can obtain nonstandard hulls of arbitrarily large cardinality.

Example 4.3. Approachable points play an important role in the study of nonstandard hulls. We define the concept as follows.

Let $(M, d)$ be a standard metric space. A point $x \in M$ is approachable if for every standard $\epsilon> 0$ there is a standard $a \in M$ such that $d(x, a) \le \epsilon $ .

The approachable points in $M \cap \mathbf {S}_w$ become exactly the standard points of the closure of M in its nonstandard hull $\widehat {M}_w$ . Indeed, for standard $f \in B_w$ with $f(w) = x \in M$ we have

$$ \begin{align*} x \text{ is approachable } \longleftrightarrow \forall^{\operatorname{\mathrm{st}}} \epsilon> 0 \; \exists^{\operatorname{\mathrm{st}}} a \in M\; (d(x,a) \le \epsilon) \longleftrightarrow \end{align*} $$
$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}} \epsilon> 0 \; \exists^{\operatorname{\mathrm{st}}} a \in M\; (\widehat{D}(f/E,\,a) \le \epsilon) \longleftrightarrow \forall \epsilon > 0 \; \exists a \in M\; (\widehat{D}(f/E,\,a) \le \epsilon),\end{align*} $$

where the last step is by Transfer. Thus in Example 4.1 all finite $x \in \mathbb {Q}$ are approachable and consequently all standard points in $\mathbb {R}$ are in the closure of $M = \mathbb {Q}$ . By Transfer, this is true for all points in $\mathbb {R}$ . In Example 4.2 all nonstandard points of $\mathbb {N}$ are inapproachable and therefore M is closed in $\widehat {M}_w$ .

Remark 4.4. In this and other constructions in this section we use the coding based on $\Psi $ . The advantage of this choice is that it produces spaces of functions (see in particular Section 4.4). One can use $\widetilde {\Psi }$ instead, and define, e.g., $\widetilde {B}_w =\, {}^{\operatorname {\mathrm {st}}} \{ f_w \,\mid \, f(w) \text { is a finite point of }M\}.$ The advantage here is that one gets an isomorphism of the external structure $( \mathbf {B}_w, \,\mathbf {E}_w,\, \mathbf {D}_w)$ with $( B_w \cap \mathbf {S}, \;E_w \cap \mathbf {S}, \;D_w \cap \mathbf {S})$ . It is thus immediately apparent that $((\widetilde {B}_w/ \widetilde {E}_w)\cap \mathbf {S}, \,(\widetilde {D}_w/\widetilde {E}_w) \cap \mathbf {S}) $ would be isomorphic to $(\mathbf {B}_w/\mathbf {E}_w, \,\mathbf {D}_w/\mathbf {E}_w)$ if the latter quotient could be formed in $\mathrm {BST}$ . (It can be formed in $\mathrm {HST}$ and this claim is a theorem there.) For the final result the choice of coding method does not matter.

Proposition 4.5. The structures $(\widetilde {B}_w/ \widetilde {E}_w, \,\widetilde {D}_w/\widetilde {E}_w) $ and $(B_w/ E_w, \,D_w/E_w) $ are isomorphic.

Proof For standard $f,g \in M^{\mathbb {N}}$ , $f \in B_w$ iff $f_w \in \widetilde {B}_w$ , $\langle f, g \rangle \in E_w $ iff $\langle f_w, g_w \rangle \in \widetilde {E}_w$ and $D_w(f,g) = \widetilde {D}_w(f_w,g_w)$ .

Remark 4.6. In the construction of $(\widehat {M} , \widehat {D} \,)$ one can replace $\mathbb {N}$ by any infinite set I, as long as $w \in I$ is good. If $\mathbf {S}_{w_1} \subseteq \mathbf {S}_{w_2}$ , fix a standard function $h \in I_1^{I_2}$ such that $h(w_2) = w_1$ and define the standard mapping $H: M^{I_1} \to M^{I_2}$ by $H(f) = f \circ h$ . Then H is an embedding of $(B_{w_1},\, E_{w_1},\,D_{w_1})$ into $(B_{w_2},\, E_{w_2},\,D_{w_2})$ in an obvious sense, and it factors to an isometric embedding of $(B_{w_1}/ E_{w_1}, \,D_{w_1}/E_{w_1}) $ into $(B_{w_2}/ E_{w_2}, \,D_{w_2}/E_{w_2}) $ . Given any good $w_1$ and $w_2$ , let $w = \langle w_1, w_2\rangle $ ; then both $(\widehat {M}_{w_1} , \widehat {D}_{w_1} )$ and $(\widehat {M}_{w_2} , \widehat {D}_{w_2} )$ embed into $(\widehat {M}_{w} , \widehat {D}_{w} )$ .

4.2 Completeness of the nonstandard hull

This subsection illustrates how one can work with the nonstandard hull as constructed in Section 4.1.

Theorem 4.7. $(\widehat {M}_w , \widehat {D}_w \,)$ is a complete metric space.

Proof Let $\langle F_n \, \mid \, n \in \mathbb {N} \rangle $ be a standard Cauchy sequence in $(\widehat {M} , \widehat {D}\, )$ ; we prove that it converges to some $F \in \widehat {M}$ .

Using the Axiom of Countable Choice we obtain a standard sequence $\langle f_n \, \mid \, n \in \mathbb {N} \rangle $ such that $F_n = (f_n)_E$ for all $n \in \mathbb {N}$ .

For $k \in \mathbb {N}$ let $n_k$ be the least element of $\mathbb {N}$ greater than or equal to k such that

$$ \begin{align*}\forall m,n \; \left( n_k \le n\le m \rightarrow \widehat{D}(F_n, F_m) <\tfrac{1}{ k+1}\right);\end{align*} $$

note that the sequence $\langle n_k \,\mid \, k \in \mathbb {N} \rangle $ is standard. From the definition of $\widehat {D}$ we obtain, for standard k,

$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}} m,n \; \left( n_k \le n \le m \rightarrow d(f_n(w), f_m(w)) < \tfrac{1}{ k+1}\right) .\end{align*} $$

Hence $\forall ^{\operatorname {\mathrm {st}}} k \;\exists ^{\operatorname {\mathrm {st}}} m \; \forall \ell \le k$

$$ \begin{align*}\left[\, n_{\ell} \le m \,\wedge\, \forall n \; \left( n_{\ell} \le n \le m \rightarrow d(f_n(w), f_m(w)) < \tfrac{1}{ \ell+1} \right)\right].\end{align*} $$

By Countable Idealization into w-standard sets we get

$$ \begin{align*}\exists^{\operatorname{\mathrm{st}}_{w}} m \; \forall^{\operatorname{\mathrm{st}}} k\,\left[\, n_{k} \le m \,\wedge\, \forall n \; \left( n_{k} \le n \le m \rightarrow d(f_n(w), f_m(w)) < \tfrac{1}{ k+1} \right)\right].\end{align*} $$

Fix such an m; clearly it is unlimited. Consider the standard point $p = f_{n_0}(0)$ . We have $d(p, f_m(w)) \le d(f_{n_0}(0), f_{n_0}(w) ) + d(f_{n_0}(w), f_m(w))$ . The first term on the right side of the inequality is limited because $f_{n_0} $ is standard and $f_{n_0} \in B$ , and the second term is $< 1$ (take $k = 0$ ). Hence $f_m(w)$ is a finite element of M.

We note that $f_m(w)$ is w-standard; hence there is a standard function $f{\kern-1.5pt}\in{\kern-1.5pt} M^{\mathbb {N}}$ such that $f(w) {\kern-1.5pt}={\kern-1.5pt} f_m(w) $ . It follows that $f {\kern-1.5pt}\in{\kern-1.5pt} B$ . We let $F {\kern-1.5pt}={\kern-1.5pt} f_E {\kern-1.5pt}\in{\kern-1.5pt} \widehat {M}$ .

We have that, for all standard k,

$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}} n \; \left( n_{k} \le n \rightarrow d(f_n(w), f(w)) < \tfrac{1}{ k+1} \right),\end{align*} $$

and hence

$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}} n \; \left( n_{k} \le n \rightarrow \widehat{D}(F_n, F) \le \tfrac{1}{ k+1} \right).\end{align*} $$

This shows that the sequence $\langle F_n \, \mid \, n \in \mathbb {N} \rangle $ converges to F.

The proof of Theorem 4.2 goes through for any infinite set I in place of $\mathbb {N}$ , as long as $w \in I$ is good.

4.3 Nonstandard hulls of standard uniform spaces

We generalize the construction of nonstandard hulls in $\mathrm {BST}$ to uniform spaces.

Let $(M, \Delta )$ be a standard uniform space. That is, M is a standard set and $\Delta $ is a standard family of pseudo-metrics on M which endows M with a Hausdorff uniform structure. In a uniform space, $x \in M$ is finite if for all standard $d \in \Delta $ , $d(x,p)$ is limited for some (equivalently: for all) standard $p \in M$ . Points x and $ y $ are infinitely close if $d(x,y) \simeq 0$ for all standard $d \in \Delta $ .

We fix a standard infinite set I and a $w \in I$ so that Idealization into w-standard sets over standard sets of cardinality $\le \kappa $ holds for $\kappa = \max \{ |\Delta |, \aleph _0\}$ (see Proposition 3.5). A construction of the nonstandard hull of $(M, \Delta )$ can now proceed much as in Section 4.1. We omit the subscripts indicating its dependence on w.

We let $B = \, {}^{\operatorname {\mathrm {st}}} \{ f \in M^I \,\mid \, f(w) \text { is a finite point of }M\}$ and

$$ \begin{align*} E = \, {}^{\operatorname{\mathrm{st}}} \{ \langle f, g \rangle \in B \times B \,\mid\, d(f(w), g(w)) \simeq 0 \text{ for all standard } d \in \Delta \}. \end{align*} $$

For each standard $d \in \Delta $ the standard function D on $B \times B$ , as well as $\widehat {M}$ , $\widehat {D}$ and c, are defined as in Section 4.1. Each $\widehat {D}$ is a pseudo-metric on $\widehat {M}$ . We let $\widehat {\Delta } =\, {}^{\operatorname {\mathrm {st}}} \{\widehat {D} \,\mid \, d \in \Delta \cap \mathbf {S}\}$ .

Theorem 4.8. The structure $(\widehat {M}, \widehat {\Delta })$ is a complete Hausdorff uniform space and c embeds $(M, \Delta )$ into $(\widehat {M}, \widehat {\Delta })$ .

Proof The proof follows the lines of the proof of Theorem 4.7; the main difference is that Cauchy sequences have to be replaced by Cauchy nets.

Let $\langle \Lambda , \le \rangle $ be a standard directed set and $\langle F_{\lambda } \,\mid \, \lambda \in \Lambda \rangle $ a standard Cauchy net indexed by $\Lambda $ . Using the Axiom of Choice one obtains a standard net $\langle f_{\lambda } \,\mid \, \lambda \in \Lambda \rangle $ such that $f_{\lambda } \in F_{\lambda }$ holds for all $\lambda $ ; then $\langle f_{\lambda }(w) \,\mid \, \lambda \in \Lambda \rangle $ is a w-standard net.

The Cauchy property implies that for every standard $d\in \Delta $ there is a standard sequence $\langle \lambda _k^d \,\mid \,k \in \mathbb {N} \rangle $ of elements of $\Lambda $ such that $\lambda _k^d \le \lambda _{k+1}^d $ holds for all k and

$$ \begin{align*}\forall k\,\forall \lambda, \mu \in \Lambda \; \left( \lambda_k^d \le \lambda \le \mu \rightarrow \widehat{d}(F_{\lambda}, F_{\mu}) <\tfrac{1}{ k+1}\right) .\end{align*} $$


$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}} k\,\forall^{\operatorname{\mathrm{st}}} \lambda, \mu \in \Lambda \; \left( \lambda_k^d \le \lambda \le \mu \rightarrow d(f_{\lambda}(w), f_{\mu}(w)) <\tfrac{1}{k+1}\right).\end{align*} $$

We conclude that for every standard $k \in \mathbb {N}$ , every standard finite $\Delta _0 \subseteq \Delta $ and every standard finite $\Lambda _0 \subseteq \Lambda $ there is a standard $\mu \in \Lambda $ such that for all $\ell \le k$ , all $d \in \Delta _0$ and all $\lambda \in \Lambda _0$

$$ \begin{align*}\lambda_{\ell}^d \le \mu \,\wedge\, \left( \lambda_{\ell}^d \le \lambda \rightarrow d(f_{\lambda}(w), f_{\mu}(w)) < \tfrac{1}{\ell+1} \right).\end{align*} $$

By Idealization into w-standard sets over sets of cardinality $\le \kappa = \max \{ |\Delta |, \aleph _0\}$ we get a w-standard $\mu \in \Lambda $ such that for all standard $ k$ , all standard $ d \in \Delta $ and all standard $\lambda \in \Lambda $

$$ \begin{align*}\lambda_k^d \le \mu \,\wedge\, \left( \lambda_k^d \le \lambda \rightarrow d(f_{\lambda}(w), f_{\mu}(w)) < \tfrac{1}{k+1} \right).\end{align*} $$

Fix such a $\mu $ ; as in Section 4.7 we have that $d(p, f_{\mu }(w))$ is limited for every standard $d \in \Delta $ , i.e., $f_{\mu }(w)$ is a finite point of M.

By Representability there is a standard function $f \in M^I$ such that $f(w) = f_{\mu }(w)$ . It follows that $f \in B$ . Let $F = f/E \in \widehat {M}$ .

We have that, for all standard $d \in \Delta $ and $k \in \mathbb {N}$ ,

$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}} \lambda \in \Lambda \; \left( \lambda_{k}^d \le \lambda \rightarrow d(f_{\lambda}(w), f(w)) < \tfrac{1}{k+1} \right) ,\end{align*} $$

and hence

$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}} \lambda \in \Lambda\; \left( \lambda_{k}^d \le \lambda \rightarrow \widehat{D}(F_{\lambda}, F) \le \tfrac{1}{k+1}\right).\end{align*} $$

This shows that the net $\langle F_{\lambda } \, \mid \, \lambda \in \Lambda \rangle $ converges to F.

4.4 Internal normed vector spaces

Under many circumstances the type of construction carried out in Sections 4.1 and 4.3 generalizes to internal structures. We illustrate it in the case of normed vector spaces.

Let M be an internal normed vector space over $\mathbb {R}$ . This means that M, the operations of addition $+$ on $M \times M$ and scalar multiplication $\cdot $ on $\mathbb {R} \times M$ , and the $\mathbb {R}$ -valued norm $ \|. \| $ on M, are (internal) sets and satisfy the usual properties. In order to be able to apply our coding technique we fix a standard set I and a good $w \in I$ so that the set M, the above operations, and the norm belong to $\mathbf {S}_w$ . Other parameters relevant to a particular investigation can also be made to belong to $\mathbf {S}_w$ . By Transfer in $(\mathbf {V}, \mathbf {S}_w, \in )$ , the properties of these objects that are expressible by $\in $ -formulas continue to hold in $\mathbf {S}_w$ , so we can carry out the desired construction “over $\mathbf {S}_w$ ” rather than “over $\mathbf {V}$ .”

We first describe the external construction. Let

It is clear that $\mathbf {B}_w$ is an external vector space over the external field $\mathbb {R} \cap \mathbf {S}$ and $\mathbf {E}_w$ is its subspace. Define an external equivalence relation $\approx $ on $\mathbf {B}_w$ by $x \approx y \longleftrightarrow x - y \in \mathbf {E}_w \longleftrightarrow \| x-y \| \simeq 0$ . Obviously, for $x_1, x_2, y_1, y_2 \in \mathbf {B}_w$ and $c \in \mathbb {R} \cap \mathbf {S}$ we have $x_1 \approx y_1 \,\wedge \, x_2 \approx y_2 \rightarrow x_1 + x_2 \approx y_1 + y_2$ , $c\cdot x_1 \approx c\cdot y_1$ and $\|x_1\| \simeq \|y_1\|$ . If external collections of external sets were available in $\mathrm {BST}$ , one could now form the quotient space $\mathbf {B}_w/\mathbf {E}_w$ with the norm $\|x_{\mathbf {E}_w}\| = \operatorname {\mathrm {sh}}(\|x\|)$ , which would then be an external normed vector space over the external field $\mathbb {R} \cap \mathbf {S}$ . This construction is of course not possible in $\mathrm {BST}$ directly, but the coding introduced in Section 3.2 enables us to carry it out and produce a standard normed metric space which, when viewed from the standard point of view, is (externally) isomorphic to $\mathbf {B}_w/\mathbf {E}_w$ .

It follows from Representability that there is a standard function $\langle (M_i, \,+_i, \, \cdot _i) \,\mid \, i \in I \rangle $ such that $(M_w, \,+_w, \, \cdot _w) = (M, \,+, \, \cdot )$ and, for all $i \in I$ , $(M_i, \,+_i, \, \cdot _i)$ is a vector space over $\mathbb {R}$ . Let

$$ \begin{align*}B_w &= {}^{\operatorname{\mathrm{st}}} \{f \in \Pi_{i \in I} M_i \,\mid\, \|f(w)\| \text{ is limited} \} ;\\E_w &= {}^{\operatorname{\mathrm{st}}} \{f \in \Pi_{i \in I} M_i \,\mid\, \|f(w)\| \simeq 0\}. \end{align*} $$

Addition and scalar multiplication on $B_w$ are defined pointwise:

$$ \begin{align*}(f+ g)(i) = f(i) +_i g(i) \qquad\text{and} \qquad (c \cdot f)(i) = c \cdot_i f(i)\end{align*} $$

for $f, g \in B_w$ , $c \in \mathbb {R}$ , and all $i \in I$ . By Standardization, there is a standard function $\|.\|$ such that for all standard $f \in B_w$ we have $\|f\| = \operatorname {\mathrm {sh}}(\|f(w)\|)$ .

It is routine to verify that $B_w$ is a standard vector space over $\mathbb {R}$ , $\|.\|$ is a pseudo-norm on $B_w$ , and $f \in E_w \longleftrightarrow \|f\| = 0$ . The quotient $\widehat {E}_w = B_w/E_w$ is thus a well-defined standard normed vector space. The proof given in Section 4.2 shows that $\widehat {E}_w = B_w/E_w$ is complete.

4.5 Loeb measures

Let $(\Omega , \mathcal {A}, \mu )$ be an internal finitely additive measure space, with $\Omega \subseteq O$ for a standard set O. As discussed in Section 6.3, attempts to construct its Loeb extension “over $\mathbf {V}$ ” are only partially successful in $\mathrm {BST}$ . We fix a standard set I and a good $w\in I$ so that $\Omega , \mathcal {A} , \mu $ are w-standard. By Transfer, $(\Omega , \mathcal {A}, \mu )$ is an internal finitely additive measure space in the sense of $\mathbf {S}_w$ , and we construct the Loeb extension “over $\mathbf {S}_w$ .” We usually do not indicate the dependence on the choice of w. In this example it is convenient to employ the variant of coding from Definition 3.10.

For every $X \in \mathcal {A}\cap \mathbf {S}_w$ let

(1) $$ \begin{align} [X] =\, {}^{\operatorname{\mathrm{st}}}\{ f_w \,\mid\, f \in O^I \,\wedge\, f(w) \in X \cap \mathbf{S}_w \}. \end{align} $$

If $X, X_1, X_2 \in \mathcal {A}\,\cap \,\mathbf {S}_w$ , then the equivalences $ f_w \in [X_1 \cap X_2] $ iff $f_w \in [X_1]\,\wedge \, f_w \in [X_2]$ and $f_w \in [\Omega \setminus X]$ iff $f_w\in [\Omega ] \,\wedge \,f_w \notin [X]$ hold for all standard $f \in O^I$ . By Transfer, $ [X_1 \cap X_2] = [X_1]\cap [X_2]$ and $ [\Omega \setminus X]= [\Omega ] \setminus [X]$ . Let

$$ \begin{align*}\mathcal{B} =\, {}^{\operatorname{\mathrm{st}}} \{ A \in \mathcal{P}(O^I)\,\,\mid\, A = [X] \text{ for some } X \in \mathcal{A} \cap \mathbf{S}_w\} .\end{align*} $$

Using Transfer again, it follows that $\mathcal {B}$ is a standard algebra of subsets of $[\Omega ]$ . We note that $X_1, X_2 \in \mathcal {A} \cap \mathbf {S}_w$ , $X_1 \ne X_2$ , implies $[X_1] \ne [X_2]$ , so for standard $A \in \mathcal {B}$ there is a unique $X \in \mathcal {A} \cap \mathbf {S}_w$ with $A = [X]$ .

A standard finitely additive measure m on the algebra $\mathcal {B}$ with values in the interval $[0, +\infty ]$ is determined by the requirement that for standard $A = [X] \in \mathcal {B}$

$$ \begin{align*}m(A) = \operatorname{\mathrm{sh}}(\mu(X)) .\end{align*} $$

The measure space $([\Omega ], \mathcal {B}, m)$ satisfies Carathéodory’s condition.

Lemma 4.9. If $\langle A_k \,\mid \, k \in \mathbb {N}\rangle $ is a sequence of mutually disjoint sets in $\mathcal {B}$ , $A \in \mathcal {B}$ , and $A = \bigcup _{k \in \mathbb {N}}\, A_k$ , then $m(A) = \Sigma _{k \in \mathbb {N}}\; m(A_k)$ .

Proof In view of Transfer, it suffices to prove this claim under the assumption that $\langle A_k \,\mid \, k \in \mathbb {N}\rangle $ and A are standard.

Suppose $A = [X]$ where $X \in \mathcal {A} \cap \mathbf {S}_w$ and for each standard k, $A_k = [X_k]$ where $X_k \in \mathcal {A} \cap \mathbf {S}_w$ . Clearly $X_k $ are mutually disjoint and $X_k \subseteq X$ for all standard k.

Assume that for every standard n there is a w-standard $x \in X$ such that $x \notin X_k$ holds for all $k \le n$ . By Countable Idealization into $\mathbf {S}_w$ there is a w-standard $x \in X$ such that $x \notin X_k$ holds for all standard k. By Representability, $x = g(w)$ for some standard $g \in O^I$ . Then $g_w \in A$ but $\forall ^{\operatorname {\mathrm {st}}} k \, (g_w \notin A_k)$ and, by Transfer, $\forall k \, (g_w \notin A_k)$ . This is a contradiction.

Therefore there is a standard n such that $\forall ^{\operatorname {\mathrm {st}}_w} x \in X \, \exists k \le n \, (x \in X_k)$ . It follows that $\forall ^{\operatorname {\mathrm {st}}} F \in A \, \exists k \le n \, (F \in A_k)$ . By Transfer, $ \bigcup _{k \in \mathbb {N}} \, A_k= \bigcup _{k \le n} \, A_k$ and, by finite additivity of m, $m(A) = \Sigma _{k \le n}\; m(A_k) = \Sigma _{k \in \mathbb {N}}\; m(A_k)$ .

We conclude that m can be extended to a $\sigma $ -additive measure $\overline {m}$ with values in $[0, +\infty ]$ on the $\sigma $ -algebra $\overline {\mathcal {B}}$ generated by $\mathcal {B}$ . The measure-theoretic completion of $([\Omega ], \overline {\mathcal {B}}, \overline {m})$ is the desired Loeb measure space; we denote it $([\Omega ], \widehat {\mathcal {B}}, \widehat {m})$ (of course, it depends on the choice of w). Instead of an appeal to the Carathéodory’s theorem, a direct proof along the lines of [Reference Hrbacek10] can be given; see also [Reference Albeverio, Høegh-Krohn, Fenstad and Lindstrøm1, Remark 3.1.5] and the references therein.

Remark 4.10. In order to explicate the relationship of $([\Omega ], \widehat {\mathcal {B}}, \widehat {m})$ to the usual external Loeb measure space, we first recall that $\mathbf {S}_w$ satisfies the statement that $(\Omega , \mathcal {A}, \mu )$ is a finitely additive measure space. The Loeb construction carried out over $\mathbf {S}_w$ would start with the external finitely additive measure space $(\boldsymbol {\Omega }, \boldsymbol {\mathcal {A}}, \mathbf {m} )$ , where $\boldsymbol {\Omega } = \Omega \cap \mathbf {S}_w$ , , and $ \mathbf {m}(X \cap \mathbf {S}_w) = \operatorname {\mathrm {sh}}(\mu (X))$ for $X \in \boldsymbol {\mathcal {A}}$ . This space would be extended to an external $\sigma $ -additive measure space using the Carathéodory’s theorem, and then completed. We now take $V = \Omega \cup \mathbb {R}$ , say, and note that for $X \in \mathcal {A} \cap \mathbf {S}_w$ , $[X]$ is a w-code of $X \cap \mathbf {S}_w$ , and hence $\mathcal {B}$ is a w-code for $\boldsymbol {\mathcal {A}}$ . It follows that $([\Omega ], \mathcal {B}, m)$ is a w-code for $(\boldsymbol {\Omega }, \boldsymbol {\mathcal {A}}, \mathbf {m} )$ . The above construction cannot be carried out in $\mathrm {BST}$ directly for the external measure space $(\boldsymbol {\Omega }, \boldsymbol {\mathcal {A}}, \mathbf {m} )$ , but presents no difficulties for its w-code $([\Omega ], \mathcal {B}, m)$ , a standard finitely additive measure space.

Remark 4.11. We compare Loeb measure spaces obtained from the same $(\Omega , \mathcal {A}, \mu )$ for different choices of the parameter w. We use subscripts to indicate dependence on this parameter and fix good $w \in I$ , $z \in J$ where $I,J$ are standard and $\mathbf {S}_w \subseteq \mathbf {S}_z$ .

Proposition 4.12. There is a standard embedding $\widehat {H} = \widehat {H}_{w,z}$ of $\widehat {\mathcal {B}}_w$ into $\widehat {\mathcal {B}}_z$ that preserves complements and countable unions and restricts to an isomorphism of $\mathcal {B}_w$ and $\mathcal {B}_z$ . If $m_w$ is $\sigma $ -finite, then also $\widehat {m}_w (B) = \widehat {m}_z (\widehat {H} (B) )$ for all $B \in \widehat {\mathcal {B}}_w$ .

Proof By Standardization, there is a standard function $\widehat {H}: \widehat {\mathcal {B}}_w \to \widehat {\mathcal {B}}_z$ such that $\widehat {H} (B) = \widetilde {\Psi }_z (\widetilde {{\boldsymbol\Psi}}^{-1}_w (B))$ for standard $B \in \widehat {\mathcal {B}}_w$ . For $A \in \mathcal {B}_w \cap \, \mathbf {S}$ , if $A = [X]_w$ for $X \in \mathcal {A} \cap \mathbf {S}_w$ then $\widehat {H} (A) = [X]_z \in \mathcal {B}_z\cap \mathbf {S}$ , and vice versa. This shows that $\widehat {H} $ maps $\mathcal {B}_w \cap \mathbf {S}$ onto $\mathcal {B}_z \cap \mathbf {S}$ , and hence, by Transfer, $\mathcal {B}_w$ onto $\mathcal {B}_z$ .

In Section 3.2 we point out that the coding $\widetilde {\Psi }$ preserves complements and countable unions, so the same holds for $\widehat {H}$ . We give some details for the countable unions. Let $B = \bigcup _{n \in \mathbb {N} } B_n$ , where $\langle B_n \,\mid \,n \in \mathbb {N}\rangle $ is standard and $B, B_n \in \widehat {\mathcal {B}}_w$ for all $n \in \mathbb {N}$ . Define the external set , so that $B_n = \widetilde {\Psi }_w (\mathbf {X}_n)$ . We then have $B = \widetilde {\Psi }_w(\bigcup _{n \in \mathbb {N} \cap \mathbf {S}} \mathbf {X}_n) $ , so $ \widetilde {{\boldsymbol\Psi}}^{-1}_w (B) =\bigcup _{n \in \mathbb {N} \cap \mathbf {S}} \mathbf {X}_n $ and $\widehat {H} (B) = \widetilde {\Psi }_z (\widetilde {{\boldsymbol\Psi}}^{-1}_w (B) )= \widetilde {\Psi }_z (\bigcup _{n \in \mathbb {N} \cap \mathbf {S}} \mathbf {X}_n ) = \bigcup _{n \in \mathbb {N}} \widetilde {\Psi }_z(\widetilde {{\boldsymbol\Psi}}^{-1}_w (B_n)) = \bigcup _{n \in \mathbb {N}} \widehat {H} (B_n)$ .

It is clear from the definition of $m(A)$ that $m_w(A) = m_z(\widehat {H}(A))$ holds for standard $A \in \mathcal {B}_w$ , and hence by Transfer, for all $A \in \mathcal {B}_w$ . If $m_w$ is $\sigma $ -finite, then the completed Carathéodory’s measure $\widehat {m}_w$ is uniquely determined. If $\widehat {m}_w (B) \neq \widehat {m}_z (\widehat {H} (B) )$ for some $B \in \widehat {\mathcal {B}}_w$ , then $\widehat {m}_w $ and $\widehat {m}_z \circ \widehat {H}$ would be two distinct extensions of $m_w$ from $\mathcal {B}_w$ to $\widehat {\mathcal {B}}_w $ , a contradiction.

4.6 Lebesgue measure from Loeb measure

As is well known, the Lebesgue measure can be obtained from a suitable Loeb measure.Footnote 6 We give the gist of the argument in our framework, for the interval $[0, 1]$ . More details can be found in [Reference Albeverio, Høegh-Krohn, Fenstad and Lindstrøm1] (using a model-theoretic approach).

For $n \in \mathbb {N}$ let $\mathcal {T}_n = \{ i /n \,\mid \, 0 \le i \le n\}$ . Fix a nonstandard integer $w \in \mathbb {N}$ , and let $\mathcal {T} = \mathcal {T}_{w}$ (in model-theoretic frameworks the equivalent of $\mathcal {T}$ is called “hyperfinite time line”). Let $O = [0,1]$ , $\Omega = \mathcal {T}$ , $\mathcal {A} = \mathcal {P}(\mathcal {T})$ and $\mu $ the counting measure on $\mathcal {T}$ , i.e., $\mu (X) = |X|/ |\mathcal {T}|$ for all sets $X \subseteq \mathcal {T}$ . The construction in the preceding subsection, with $I =\mathbb {N}$ , yields the Loeb measure $([\Omega ], \widehat {\mathcal {B}}, \widehat {m})$ .

For every standard $A \subseteq [0,1]$ define

$$ \begin{align*}{}^{\bullet}\!\!A =\, {}^{\operatorname{\mathrm{st}}}\{ f _w\in [\Omega] \,\mid\, f(w) \simeq c \text{ for some standard } c \in A\}.\end{align*} $$

This just means that ${}^{\bullet }\!\!A$ is a w-code for $ \operatorname {\mathrm {sh}}^{-1} (A) \cap \mathcal {T} \cap \mathbf {S}_w $ . Standard elements of the set ${}^{\bullet }\!\!A$ are those $f_w \in [\Omega ]$ whose value “at infinity” (i.e., at w) is infinitely close to a standard real in A.

Let $\mathcal {L} =\, {}^{\operatorname {\mathrm {st}}}\{ A \subseteq [0,1] \,\mid \,{}^{\bullet }\!\!A \in \widehat {\mathcal {B}} \,\}$ and let $\ell $ be the standard function on $\mathcal {L}$ determined by the requirement that $\ell (A) = \widehat {m}({}^{\bullet }\!\!A) $ for all standard $A \in \mathcal {L}$ .

Theorem 4.13. The triple $([0,1], \mathcal {L}, \ell )$ is the Lebesgue measure space on $[0,1]$ .

Proof We prove that $\mathcal {L}$ is a $\sigma $ -algebra containing all standard open intervals $(a,b) \subseteq [0,1]$ and all singletons $\{c \}$ for standard $c \in [a,b]$ . We also prove that $\ell $ is $\sigma $ -additive and $\ell ( (a,b)) = b-a$ , $\ell (\{c\}) =0$ for all standard open intervals and singletons, respectively. This implies that $\mathcal {L}$ contains all Lebesgue measurable subsets of $[0,1]$ and that $\ell $ is the Lebesgue measure for such subsets. For a proof (in the model-theoretic framework) that all sets in $\mathcal {L}$ are Lebesgue measurable see [Reference Albeverio, Høegh-Krohn, Fenstad and Lindstrøm1, Proposition 3.2.5]; it can be easily adapted to our framework.

Let $\langle A_k \,\mid \, k \in \mathbb {N}\rangle $ be a standard sequence of elements of $\mathcal {L}$ and let $A = \bigcup _{k \in \mathbb {N}} A_k$ . Then ${}^{\bullet }\!\!A_k\in \widehat {\mathcal {B}} $ holds for all standard $k \in \mathbb {N}$ , and we obtain that ${}^{\bullet }\!\!A = \bigcup _{k \in \mathbb {N}} {}^{\bullet }\!\!A_k$ , because if $f(w) \simeq a $ for some standard $a \in A$ , then $a \in A_k$ for some standard $k \in \mathbb {N}$ by Transfer. As $\widehat {\mathcal {B}}$ is a $\sigma $ -algebra, ${}^{\bullet }\!\!A \in \widehat {\mathcal {B}}$ and we conclude that $A \in \mathcal {L}$ . Furthermore, $\ell (A) =\widehat {m}({}^{\bullet }\!\!A)= \Sigma _{k \in \mathbb {N}}\; \widehat {m}({}^{\bullet }\!\!A_k) = \Sigma _{k \in \mathbb {N}}\;\ell (A_k)$ , establishing $\sigma $ -additivity of $\ell $ .

Given a standard open interval $(a, b) \subseteq [0,1]$ , we let $X_{a,b} = \mathcal {T} \cap (a,b)$ . We have $[X_{a,b}] \in \mathcal {B}$ and $m([X_{a,b}]) = \operatorname {\mathrm {sh}}(\mu ( X_{a,b} ) )= b-a$ .

Let $A = (a,b) $ ; it remains to observe that, for standard $f_w \in [\Omega ]$ , $f_w \in {}^{\bullet }\!\!A$ iff $f_w \in [X_{a + 1/m, \;b - 1/m} ]$ for some standard $m \in \mathbb {N} \setminus \{0\}$ . Standardization gives the function $\langle A_m\,\mid \, m \in \mathbb {N}\setminus \{0\}\rangle $ where $A_m = [X_{a + 1/m, \;b - 1/m} ]$ for standard m, and Transfer implies ${}^{\bullet }\!\!A = \bigcup _{m \in \mathbb {N}\setminus \{0\}} A_m$ . It follows that ${}^{\bullet }\!\!A \in \overline {\mathcal {B}}$ because the latter is a $\sigma $ -algebra, and that $\overline {m}(\overline {A}) = b - a$ because $\overline {m}$ is a $\sigma $ -additive extension of m. Hence $A \in \mathcal {L}$ and $\ell (A) = b - a$ .

The argument for $A = \{c\}$ is similar, using the fact that ${}^{\bullet }\!\!A = \bigcap _{m \in \mathbb {N}\setminus \{0\}} A_m$ with $A_m = [X_{c- 1/m,\, c + 1/m} ]$ for standard m.

4.7 Neutrices and external numbers

This is another application of nonstandard analysis that extensively uses external sets; see [Reference Dinis and van den Berg6].

A neutrix is a convex additive subgroup of $\mathbb {R}$ . With the exception of $\{0\}$ and $\mathbb {R}$ , neutrices are externals sets; the monad $\mathbf {M}(0)$ and the galaxy $\mathbf {G}(0)$ are nontrivial examples. An external number is an algebraic sum of a real number and a neutrix. Addition and multiplication of external numbers are defined by the Minkowski operations. Let $\mathcal {N}$ denote the collection of all neutrices and $\mathcal {E}$ denote the collection of all external numbers. Even in $\mathrm {HST}$ , $\mathcal {N}$ and $\mathcal {E}$ are (definable) proper classes of external sets; they are “too large” to be external sets (see [Reference Kanovei and Reeken13]).

This difficulty can be remedied by relativizing these concepts to $\mathbf {S}_w$ (for good w). In fact, and . The collections $\mathcal {N}_w$ and $\mathcal {E}_w $ are external sets (of external sets) in $\mathrm {HST}$ .

As the natural embedding of $\mathbb {R} \cap \mathbf {S}_w$ into $\mathcal {E}_w$ given by $r \mapsto r+\{0\}$ is crucial for applications of these concepts, it may be best for most purposes to avoid coding as much as possible. For example, the study of algebraic properties of operations $+$ and $\times $ on