Let $f\in L^2(0,1)$ and $g\in L^2(0,1)$. Is any function that lies point-wise between $f$ and $g$ going to be Lebesgue measurable, i.e. $$f(x)\le h(x)\le g(x), \quad \forall x\in [0,1]$$ then is $h(x)$ Lebesgue measurable?
2026-03-27 02:01:14.1774576874
Will this function be measurable
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It's glaringly false. Pick $f=0$, $g=1$ and $h=\chi_{\text{Vitali set}}$.