Given a simplex S and a positive function δ on S, we show that there is a simplicial subdivision of S such that the diameter of each subdividing simplex is smaller that δ evaluated at some of its vertices.

If П is a k-dimensional vector subspace of Rn and E is a subset of Rn, let projп(E) denote the orthogonal projection of E onto П. Marstrand [8] and Kaufman [6] have developed results on the Hausdorff dimension and measure of projп(E) in terms of the dimension of E, leading to the very general theory of Mattila [11]. In particular, Mattila shows that if the Hausdorff dimension dim E of the Souslin set E is greater than k, then projп(E) has positive k-dimensional Lebesgue measure for almost all П ∈ Gn, k (in the sense of the usual normalized invariant measure on the Grassmann manifold Gn, k of k-dimensional subspaces of Rn).

Let (X, ℱ, μ) be a topological measure space with X a completely regular Hausdorff space and ℱ the σ-algebra of all μ-measurable sets, containing all the Baire sets of X. Consider the following two conditions on (X, ℱ, μ).

This note contains characterizations of those sigma-fields for which sigma-finiteness is a necessary condition in the Radon-Nikodym Theorem.

Our purpose is to consider those σ-fields for which σ-finiteness is a necessary condition in the Radon–Nikodym Theorem. We first prove a measure theoretic equivalence in the general case, and then use this to obtain an algebraic characterization in the case when the σ-field is the Borel field of a locally compact separable metric space. For undefined terminology we refer the reader to [1] for measure theoretic and [2] for algebraic properties.

By a measure, we mean a countably additive function from σ-field of sets or a Boolean σ-algebra into the non-negative extended real numbers. We will say that a measure μ on a σ-field of sets Σ is RN provided each μ-continuous finite measure on Σ has a Radon–Nikodym derivative in L1(μ).

Let X be a weakly complete proper cone contained in a weak space E and h(E) the Riesz space generated by the continuous linear forms on E. A positive conical measure μ on X is a positive linear form on h(E)|x. G. Choquet has proved μ is a Daniell integral on E when E is weakly complete, but μ is not generally a Daniell integral on X. However we give an integration theory for functions on X and compare this theory with the classical Daniell theory. The case where μ is maximal in the sense of G. Choquet is remarkable.

There are at least two indices used to measure the size of bounded sets of ℝn of zero measure—Hausdorff dimension (see [4] for a definition), and the density index [7].

§1. Introduction and notation. In [1] and [2], Besicovitch demonstrated that there exist plane sets of measure zero containing line segments (and indeed entire lines) in all directions in the plane. It is natural to ask about the existence of analogous sets in Euclidean spaces of higher dimensions, and in [3] we defined an (n, k)-Besicovitch set to be a subset A of Rn, of n-dimensional Lebesgue measure zero, such that for each k-dimensional subspace Π of Rn, some translate of Π intersects A in a set of positive k-dimensional measure. (Thus Besicovitch's original constructions were for (2,1)-Besicovitch sets.) Recently, Marstrand [5] has shown (by approximating to sets by unions of cubes) that no (3, 2)-Besicovitch sets exist, and simultaneously the author [3] proved using Fourier transform methods that (n, k)- Bsicovitch sets cannot exist if k > ½n.

The setting is a compact Hausfroff space ω. The notion of a Walls class of subsets of Ω is defined via strange axioms—axioms whose justification rests with examples such as the collection of regular open sets or the range of a strong lifting. Avarient of Rosenthal' famous lwmma which applies directly to Banach space-valued measures is esablished, and it is used to obtain, in elementary fashion, the following two uniform boundedness principles: (1)The Nikodym Boundedness Theorem. If K is a family of regular Borel vector measures on Ω which is point-wise bounded on every set of a fixed Wells class, then K is uniformly bounded. (2)The Nikodym Covergence Theorem. If {μn} is a sequence of regular Borel vector measures on Ω which is converguent on every set of a fixed Wells class, then the μn are uniformly countably additive, the sequence {μn} is convergent on every Borel subset of Ω and the pointwise limit constitutes a regular Borel measure.

We denote by S the unit sphere in ℝ3, and µ is the rotationally invariant measure, generalizing surface area on S; thus µS = 4π. We identify directions (or unit vectors) in ℝ3 with points on S, and prove the following:

Theorem 1. If E is a subset of ℝ3 of Lebesgue measure zero, then for µ almost all directions α, every plane normal to α intersects E in a set of plane measure zero.

Suppose μ and ν are Borel measures on locally compact spaces X and Y, respectively. A product measure λ can be defined on the Borel sets of X x Y by the formula λ(M) = ∫ν(Mx) dμ, provided that vertical cross section measure ν(Mx) is a measurable function in x. Conditions are summarized for ν(Mx) to be measurable as a function in x, and examples are given in which the function ν(Mx) is not measurable. It is shown that a dense, countably compact set fails to be a Borel set if it contains no nonempty zero set.

In [1], [2] Besicovitch showed that it is possible to translate each straight line in the plane so that the union of all the translates has zero plane measure. More recently Besicovitch and Rado [3] and independently Kinney [12] showed that the same can be done with arcs of circles instead of straight lines (see also Davies [6]). Allowing rotations as well as translations, Ward [18] showed that all plane polygonal curves can be “packed” thus (allowing overlapping) into zero plane measure, and then Davies [7], making use of Besicovitch's construction, showed translations alone to be sufficient, although these papers in fact contained stronger results concerning Hausdorff measure; the results were further generalized in [16]. The question has naturally been asked whether the class of all plane rectifiable curves can be packed by isometries (translations and rotations) into zero plane measure, but a special case of the main theorem of the present paper shows that this is impossible. The corresponding question remains open for the much smaller class of algebraic curves, or even conies.

In a typical counter-example construction in geometric measure theory, starting from some initial set one obtains by successive reductions a decreasing sequence of sets Fn, whose intersection has some required property; it is desired that ∩ Fn shall have large Hausdorf F dimension. It has long been known that this can often be accomplished by making each Fn+1 sufficiently “dense” in Fn. Our first theorem expresses this intuitive idea in a precise form that we believe to be both new and potentially useful, if only for simplifying the exposition in such cases. Our second theorem uses just such a construction to solve the problem that originally stimulated this work: can a Borel set in ℝk have Hausdorff dimension k and yet for continuum-many directions in every angle have at most one point on each line in that direction? The set of such directions must have measure zero, since in fact in almost all directions there are lines that meet the Borel set (of dimension k) in a set of dimension 1: this can easily be deduced from Theorem 6.6 of Mattila [5], which generalized Marstrand's result [4] for the case k = 2.

We exhibit (§2) an example of §a compact Hausdorff space supporting a Radon probability measure μ and a continuous map ø : X → I, when I is the closed unit interval, for which the image measure ø(μ) is Lebesgue measur m with the properties:

(i) there exists an open set G ⊂ X for which ø(G) is not m-measurable;

(ii) μ is a non-atomic non-completion regular measure;

(iii) the measure algebras (X, μ) and (I, m) are isomorphic but for no choice c sets B ⊂ X, B′ ⊂ I of measure zero are and homeomorphic

(iv) there exists a selection p : I → X (i.e. p(t) ∊ ø−1(t) for all t ∊ I) which i Borel m-measurable, but there is no Lusin m-measurable selection.

A certain natural extension B of the Borel σ-algebra is studied in generalized weakly θ-refinable spaces. It is shown that a set belongs to B whenever it belongs to B locally. From this it is derived that if ℵωα is more complicated than aunion of less than ℵα weakly θ-refinable subspaces.

A complete characterization of Boolean algebras which admit nonatomic charges (i.e. finitely additive measures) is obtained. This also gives rise to a characterization of superatomic Boolean algebras. We also consider the problem of denseness of the set of all nonatomic charges in the space of all charges on a given Boolean algebra, equipped with a suitable topology.

Let μ be a Borel measure on a completely regular space X, and denote by ℱ the σ-algebra of all μ*-measurable subsets of X. Suppose that, as an abstract measure space, (X, ℱ, μ) is isomorphism mod zero with the standard Lebesgue space (I, ℒ, m) via an isomorphism φ : X → I. In this note we attempt to answer the following question: Under what conditions can the isomorphism φ be chosen to be a homeomorphism mod zero? When X is compact, the existence of such a homeomorphism was established in [3, §4] under the assumption of uniform regularity of μ. Whether or not the result can be established without this assumption, was posed as an open question there. Here, we give necessary and sufficient conditions for the existence of the above homeomorphism, together with various examples showing, among other things, that the assumption of uniform regularity used in [3, §4] cannot be dropped.

Davies and Rogers [5] constructed a compact metric space Ω which is singular for a certain Hausdorff measure μh, in the sense that all subsets of Ω have μh-measure zero or infinity and μh(Ω) = ∞. (For a further study of this example see Boardman [3]). The interest lies in its extremely good descriptive character, which was lacking in the earlier examples given by Besicovitch [2] (a plane set singular for linear measure) and Choquet [4] (a plane set singular for any Hausdorff measure for which a segment has positive measure).

Let G be any enumerable subset of the positive real numbers, with infinity as its only limit point. The purpose of this paper is to give a construction for a Lebesgue measurable set E ⊂ R+, with the following properties:

Definitions. We say that h is a Hausdorff measure function if it satisfies the following conditions:

(i) for all x > 0,

(ii) h(x) → 0 as x → 0,

(iii) h(x) is monotonic increasing.

Our result complements an interesting result of Roy O. Davies [1]; we assume familiarity with his paper. We use the details of the construction that he uses to prove his Theorem II.