Lineability within Peano Curves, Martingales, and Integral Theory

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.