Lineability within Peano Curves, Martingales, and Integral Theory

Artur Bartoszewicz,Szymon Głąb,Marek Bienias

doi:10.1155/2018/9762491

Abstract

This paper is devoted to give several improvements of some known facts in lineability approach. In particular, we prove that (i) the set of continuous mappings from the unit interval onto the unit square contains a closed,c-semigroupable convex subset, (ii) the set of pointwise convergent martingales(Xn)n∈NwithEXn→∞isc-lineable, (iii) the set of martingales converging in measure but not almost surely isc-lineable, (iv) the set of sequences(Xn)n∈Nof independent random variables, withEXn=0,∑n=1∞var Xn=∞, and the property that(X1+⋯+Xn)n∈Nis almost surely convergent, isc-lineable, (v) the set of bounded functionsf:[0,1]×[0,1]→Rfor which the assertion of Fubini’s Theorem does not hold is consistent withZFC 1-lineable (it is not 2-lineable), (vi) the set of unbounded functionsf:[0,1]×[0,1]→Rfor which the assertion of Fubini’s Theorem does not hold (with infinite integral allowed) isc-lineable but notc+-lineable.

Highlights

We prove that (i) the set of continuous mappings from the unit interval onto the unit square contains a closed, c-semigroupable convex subset, (ii) the set of pointwise convergent martingales (Xn)n∈N with E|Xn| 󳨀→ ∞ is c-lineable, (iii) the set of martingales converging in measure but not almost surely is c-lineable, ∑∞ n=1 varXn = ∞, and the property that
We start with a standard introduction to the idea of lineability
For more than a decade many mathematicians have been looking at the largeness of some sets by constructing algebraic structures inside them

Summary

Introduction

We start with a standard introduction to the idea of lineability For more than a decade many mathematicians have been looking at the largeness of some sets by constructing algebraic structures inside them. Perez-Garcıa, and J.B. Seoane-Sepulveda, let us recall the following notion. (2) Let L be a semigroup and A ⊆ L. This paper is devoted to the improvements of some known facts: in semigroupability (Section 2) and lineability in probability theory (Section 3). Along this paper we shall use standard set theoretical notation and notions.

Peano Curves and Semigroupability

Lineability within Martingales Theory

Kolmogorov’s Theorem

Failure of Fubini’s Theorem