On Extremal Problems for Pairs of Uniformly Distributed Sequences and Integrals with Respect to Copula Measures

Motivated by the maximal average distance of uniformly distributed sequences we consider some extremal problems for functionals of type {\mu _C} \mapsto \int_0^1 {{{\int_0^1 {Fd} }_\mu }_C,} where µC is a copula measure and F is a Riemann integrable function on [0, 1]2 of a specific type. Such problems have been considered in [4] and are of interest in the study of limit points of two uniformly distributed sequences.

ALMOST RIEMANN INTEGRABLE FUNCTIONS

An arbitrary function $f$ on a bounded interval $[a,b]$ is termed an almost $R$-integrable function if there exists a Riemann integrable function $g$ such that $f =g$ a.e. In this note a characterization of the class of almost $R$-integrable functions is obtained.

Mean Value Theorem for the m-Integral of Dinculeanu

The classical mean value theorem asserts that if f is a real, bounded, Riemann integrable function defined on a finite real interval a≤t≤b, then , where infa≤t≤bf(t)≤y0supa≤t≤bf(t). The extensions of Choquet [3], Price [15], and of this paper generalize the fact that y0 belongs to the closure of the convex hull of f([a, b]). The version of Choquet ([3, p. 38]) applies to a continuous function on a compact interval with values in a Banach space; that of Price ([15, p. 24]) applies to a bilinear integral of a special type containing the Birkhoff integral [2]. The m-integral of Dinculeanu [6] (specialization of Bartle's *- integral [1]) leaves intact the Lebesgue dominated convergence theorem and is strong enough to support an extended development. The paper is organized as follows: the object of §2 is to express the integral of a bounded m-integrable function as a limit of Riemann sums; §3 gives Price's generalization of "convex hull" [15]; the theorem of the paper is established in §4; §5 gives applications to vector differentiation which, for continuously differentiable functions, contain results of Dieudonné [5] and McLeod [13].

A Convergence Theorem for Certain Riemann Sums

For a Riemann integrable function f on the interval [0,1], letand consider the Riemann sums