So from what I've been in reading, if f is a homeomorphism then f does not preserve Lebesgue measurability for some E, i.e, E is measurable iff f(E) is measurable.
Does this change if the inverse of f sends Lebesgue Zero Sets to Lebesgue Zero Sets?
2026-04-26 09:24:28.1777195468
Preserving Lebesgue Measurability
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No. Let $c(x)$ denote the Cantor function on $[0,1]$ and consider $f: [0,1] \to [0,2]$ given by $f(x) = c(x) + x$.