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Let $f:\mathbb R^d\longrightarrow\mathbb R$ measurable and $E\subset \mathbb R^d$ measurable, does $f(E)$ measurable ?

I proved that if $f$ is a bijection, it hold since if $g$ is the reciprocal of $f$, then $g$ is mesurable and $f(E)=g^{-1}(E)$, therefore $f(E)$ is measurable.

idm
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  • $ f (E) = E $ does not follow from $ f $ being a bijection. – littleO Oct 23 '15 at 11:10
  • Please look here http://math.stackexchange.com/questions/530280/image-of-measurable-set-under-continuous-inverse-function-is-always-measurable –  Oct 23 '15 at 11:22

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