Some theorems on the existence of continuous real-valued functions on a topological space (for example, insertion, extension, and separation theorems) can be proved without involving uncountable unions of open sets. In particular, it is shown that well-known characterizations of normality (for example the Katětov-Tong insertion theorem, the Tietze extension theorem, Urysohn's lemma) are characterizations of normal σ-rings. Likewise, similar theorems about extremally disconnected spaces are true for σ-rings of a certain type. This σ-ring approach leads to general results on the existence of functions of class α.

In this paper we shall be concerned with the growth of functions in the class ℳp, where we write f ∈ ℳp 1 <p<∞, if:

(i) f is absolutely continuous on bounded subintervals of [0, ∞]);

(ii) f(0) = 0; and

(iii) .

Let X(1) ≦ X(2) ≦ ·· ·≦ X(N(t)) be the order statistics of the first N(t) elements from a sequence of independent identically distributed random variables, where {N(t); t ≧ 0} is a renewal counting process independent of the sequence of X's. We give a complete description of the asymptotic distribution of sums made from the top kt extreme values, for any sequence kt such that kt → ∞, kt/t → 0 as t → ∞. We discuss applications to reinsurance policies based on large claims.

Simple necessary and sufficient conditions for a function to be concave in terms of its shifted Laplace transform are given. As an application of this result, we show that the expected local time at zero of a reflected Lévy process with no negative jumps, starting from the origin, is a concave function of the time variable. A special case is the expected cumulative idle time in an M/G/1 queue. An immediate corollary is the concavity of the expected value of the reflected Lévy process itself. A special case is the virtual waiting time in an M/G/1 queue.

This paper considers analogs of results on integral operators studied by Hörmander. Using the sharp function introduced by Fefferman and Stein, we prove weighted norm inequalities on kernel operators which map an Lp space into an Lq space, with q not equal to p. The techniques recover known results about fractional integral operators and apply to multiplier operators which satisfy a generalization of the Hörmander multiplier condition.

Ramsey's theorem implies that every function f:0, 1 ℝ isconvex or concave on an infinite set. We show that there is an upper semicontinuous function which is not convex or concave on any uncountable set. We investigate those functions which are not convex on any r element set (r ). A typical result: if f is bounded from below and is not convex on any infiniteset then there exists an interval on which the graph of f can be covered by the graphs of countably many strictly concave functions.

A trigonometric series has “small gaps” if the difference of the orders of successive terms is bounded below by a number exceeding one. Wiener, Ingham and others have shown that if a function represented by such a series exhibits a certain behavior on a large enough subinterval I, this will have consequences for the behavior of the function on the whole circle group. Here we show that the assumption that f is in any one of various classes of functions of generalized bounded variation on I implies that the appropriate order condition holds for the magnitude of the Fourier coefficients. A generalized bounded variation condition coupled with a Zygmundtype condition on the modulus of continuity of the restriction of the function to I implies absolute convergence of the Fourier series.

The main aim of this note is the proof of the following

Let −∞ ≤ a > b ≤ ∞ and let A ⊂ (a, b) be a measurable set such that λ((a, b)\A) = 0, where λ denotes Lebesgue measure on ℝ. Let f: A→ℝ be a measurable and midconvex function, i.e.

whenever. Then there exists a convex functionsuch that.

On compact oriented differentiable manifolds, we define a well behaved Riemann type integral which coincides with the Lebesgue integral on nonnegative functions, and such that the exterior derivative of a differentiable (not necessarily continuously) exterior form is always integrable and the Stokes formula holds.

We say that a function has locally small Riemann sums on an interval if for each point x in the interval, and for each positive number ε, all sufficiently fine partitions of intervals lying in neighborhood of x but not containing x have Riemann sums of absolute value less than ε. The main result is then as the title states. We use the generalized Riemann approach to Perron integration, assuming that functions are measurable only to insure that conditions involving the positive and negative parts of the functions are satisfied.

We present a systematic and self-contained exposition of the generalized Riemann integral in a locally compact Hausdorff space, and we show that it is equivalent to the Perron and variational integrals. We also give a necessary and sufficient condition for its equivalence to the Lebesgue integral with respect to a suitably chosen measure.

A maximality principle on quasi-ordered pseudo-metric spaces is used to obtain a number of Lipschitz attraction results for non-semigroup evolution processes with respect to time-dependent families. As particular cases, a multivalued version of Dieudonné's means value theorem and the Kirk-Ray lipschitzianness test are derived.

The special Denjoy-Bochner integral (the D*B-integral) which are generalisations of Lebesgue-Bochner integral are discussed in [7, 6, 5]. Just as the concept of numerical almost periodicity was extended by Burkill [3] to numerically valued D*- or D-integrable function, we extend the concept of almost periodicity for Banach valued function to Banach valued D*B-integrable function. For this purpose we introduce as in [3] a distance in the space of all D*B-integrable functions with respect to which the D*B-almost periodicity is defined. It is shown that the D*B-almost periodicity shares many of the known properties of the almost periodic Banach valued function [1, 4].

We introduce the notion of functions of bounded proximal variation and the notion of orderly connected topology on the real line. Using these notions, we define in a novel way an integral of Perron type, including virtually all the known integrals of Perron and Denjoy types and admitting mean value theorems and integration by parts and the analog of Marcinkiewicz theorem for the ordinary Perron integral.

If two functions of a real variable are integrable over two intervals, say of t, τ, respectively, then the product of the two functions should be integrable over the rectangular product of the two intervals of t and τ. For the Lebesgue integral, definable using non-negative functions alone, the proof is easy. For non-absolute integrals such as the Perron, Çesàro-Perron, and Marcinkiewicz-Zygmund integrals we have difficulties since the functions cannot be assumed non-negative. But the present paper gives a proof.

This paper defines descriptive, Riemann, and constructive integrals equivalent to the approximately continuous integral of Burkill.

Two index laws for fractional integrals and derivatives, which have been extensively studied by E. R. Love, are shown to be special cases of an index law for general powers of certain differential operators, by means of the theory developed in a previous paper. Discussion of the two index laws, which are rather different in appearance, can thus be unified.

Under the standard assumptions, the formula for integration by parts is obtained by a straightforward calculation.

We derive some specific inequalities involving absolutely continuous functions and relate them to a norm inequality arising from Banach algebras of functions having bounded k th variation.

A classical theorem states that any open set on the real line is a countable union of disjoint open invervals. Here the numerical content of this theorem is investigated with the methods of constructive topology.