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In this paper (PDF) Sierpinski constructed a non-measurable set from an ultrafilter on $\mathbb{N}$. The existence of ultrafilters on $\mathbb{N}$ is weaker than the Axiom of Choice.

Construct a Cantor set of positive measure in much the same way as you make the `standard' Cantor set but make sure the lengths of the deleted intervals add up to 1/2, say. Let $U$ be the union of the …

Here is an earlier effort of Sierpiński: Sur une propriété de la décomposition de M. Vitali, Mathematica 3, 30-32 (1930). He took "Vitali's Decomposition", that is, the family of cosets of $\mathbb{Q} …

Yes. Enumerate the set of all finite $0$-$1$-sequences as $\langle\sigma_n:n\in\omega\rangle$ such that each sequence is listed infinitely often. Define $s$ to be the sequence that starts with $a_0$ c …

I don't know if this is the right generalization but here is an article on Non-Archimedean probability by Benci et al (Milan J. Math. 81 (2013), no. 1, 121–151). The version of the real line used ther …

If the interior of $X$ were nonempty there would be nonempty open sets $U$ and $V$ in $L^1$ such that $U\times V\subseteq X$. But then $U\subseteq A$ shows that $A$ would have nonempty interior. Note …

I take it that the normal, self-adjoint functions are assumed to be continuous on their domains. In that case the functions $f+g$ and $f\cdot g$ are continuous on the dense open set $O=X\setminus(Z_f\ …