Suppose we have a diffeomorphism $\varphi: U \rightarrow V$ where U and V are open sets of $\mathbb{R}^n$. Can we show that the image of a measurable set A is again measurable. I know that this is the case for Borel-sets but can we show it for Lebesgue-sets? If so how can we show it?
2026-02-23 01:01:04.1771808464
Diffeomorphisms and measurable sets
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Since Lebesgue measurable sets are sets which differ from a Borel set by a subset of a Borel null set, it suffices to show that the image of a Borel null set is null. This follows immediately from the change-of-variables formula: if $E\subset U$ is a Borel null set, then $$\int_V 1_{\varphi(E)}=\int_U 1_E |\det d\varphi|=0$$ so $\varphi(E)$ is null.