Lineability within Peano curves, martingales, and integral theory (Q1733834)

Summary: This paper is devoted to give several improvements of some known facts in the lineability approach. In particular, we prove that (i) the set of continuous mappings from the unit interval onto the unit square contains a closed, \(\mathfrak{c}\)-semigroupable convex subset, (ii) the set of pointwise convergent martingales \((X_n)_{n \in \mathbb{N}}\) with \(E |X_n| \rightarrow {\infty}\) is \(\mathfrak{c}\)-lineable, (iii) the set of martingales converging in measure but not almost surely is \(\mathfrak{c}\)-lineable, (iv) the set of sequences \((X_n)_{n \in \mathbb{N}}\) of independent random variables, with \(E X_n = 0\), \(\sum_{n = 1}^{\infty} \operatorname{var} \; X_n = \infty\), and the property that \((X_1 + \cdots + X_n)_{n \in \mathbb{N}}\) is almost surely convergent, is \(\mathfrak{c}\)-lineable, (v) the set of bounded functions \(f : [0,1] \times [0,1] \rightarrow \mathbb{R}\) for which the assertion of Fubini's theorem does not hold is consistently with ZFC \(1\)-lineable (it is not 2-lineable), (vi) the set of unbounded functions \(f : [0,1] \times [0,1] \rightarrow \mathbb{R}\) for which the assertion of Fubini's theorem does not hold (with infinite integral allowed) is \(\mathfrak{c}\)-lineable but not \(\mathfrak{c}^+\)-lineable.