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We introduce Riesz potentials for Lebesgue non-measurable functions by taking the integrals in the sense of Choquet with respect to Hausdorff content and prove boundedness results for these operators. Some earlier results are recovered or extended now using integrals taken in the sense of Choquet with respect to Hausdorff content. Some earlier results also for maximal operators are considered, but now for Lebesgue non-measurable functions.

We introduce a notion of strong algebrability of subsets of linear algebras. Our main results are the following. The set of all sequences from c0c_0 which are not summable with any power is densely strongly c\\mathfrak {c}–algebrable. The set of all sequences in l∞l^\\infty whose sets of limit points are homeomorphic to the Cantor set is comeager and strongly c\\mathfrak {c}-algebrable. The set of all non-measurable functions from RR\\mathbb {R}^{\\mathbb {R}} is 2c2^\\mathfrak {c}–algebrable. These results complete several by other authors, within the modern context of lineability.

A connection between the continuous translation equation and the Jordan non-measurable continuous functions is given.

We show that there exists an infinite dimensional vector space every non-zero element of which is a non-measurable function. Moreover, this vector space can be chosen to be closed and to have dimension β for any cardinality β. Some techniques involving measure theory and density characters of Banach spaces are used.[PUBLICATION ABSTRACT]

It is well known, that for the sums of i.i.d. random variables we have Sn/n → 0 a.s. iff ∑∞n=1 1/nP(|Sn| > nε) < ∞ holds for all ε > 0 (Spitzer's SLLN). The result is also known in separable Banach spaces. It will be shown, that this also holds in nonseparable (= not necessarily separable) Banach spaces without any measurability assumption. In the theory of empirical processes this gives a characterization of Glivenko-Cantelli classes.

Comparisons are made between the expected returns using measurable and non-measurable stop rules in discrete-time stopping problems. In the independent case, a natural sufficient condition (\"preservation of independence\") is found for the expected return of every bounded non-measurable stopping function to be equal to that of a measurable one, and for that of every unbounded non-measurable stopping function to be arbitrarily close to that of a measurable one. For non-negative and for uniformly-bounded independent random variables, universal sharp bounds are found for the advantage of using non-measurable stopping functions over using measurable ones. Partial results for the dependent case are obtained.

The goal of this article is to show Fubini’s theorem for non-negative or non-positive measurable functions [ ], [ ], [ ], using the Mizar system [ ], [ ]. We formalized Fubini’s theorem in our previous article [ ], but in that case we showed the Fubini’s theorem for measurable sets and it was not enough as the integral does not appear explicitly. On the other hand, the theorems obtained in this paper are more general and it can be easily extended to a general integrable function. Furthermore, it also can be easy to extend to functional space [ ]. It should be mentioned also that Hölzl and Heller [ ] have introduced the Lebesgue integration theory in Isabelle/HOL and have proved Fubini’s theorem there.

Estimates for the measure of noncompactness of linear integral operators of vector functions in ideal spaces are obtained. When the kernel function is compact, no additional uniformity or measurability hypotheses are needed; however, noncompact nonmeasurable kernel functions are also treated.

This chapter discusses basic mathematical concepts, point‐set concepts, set and measure functions, normed linear spaces, and integration, used in Stochastic differential equations. It chapter defines a space as a type of master or universal set, which is the context in which discussions of sets occur, and looks at inclusion symbols, specifically considering element inclusion. Given two nonempty sets X and Y, a single valued function or point‐to‐point mapping f: X → Y is a rule or law of correspondence that associates with point x ∈ X a unique point y ∈ Y. The chapter explores some of the salient features of real numbers and defines the limit superior and limit inferior of a sequence of real numbers. It defines the integral of a non‐negative simple function. The chapter then defines the integral of a non‐negative measurable function via an approximation by simple functions.