In my course it's written that if $f$ measurable and $g$ continuous then $g\circ f$ is measurable. But is there example of functions $f$ and $g$ s.t. $f$ continuous, $g$ measurable and $g\circ f$ not measurable ?
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Yes, see the answer here. – Jun 16 '18 at 16:45
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@nicomezi You are confusing the general concept of measurability with the special case of real-valued functions. In general $f:(X,\mathcal E)\to (Y,\mathcal F)$ is measurable if and only if $f^{-1}[F]\in\mathcal E$ for all $F\in\mathcal F$ (and here composition of measurable is measurable). If $f:\Bbb R^n\to\Bbb R$, we say that $f$ is measurable if $f$ is measurable as a function $(\Bbb R^n,\operatorname{Lebesgue})\to (\Bbb R,\operatorname{Borel})$. This means that, for $n=1$, $f\circ g$ is measurable if $f$ is Borel and $g$ is measurable, but not necessarily if $f$ is measurable. – Jun 16 '18 at 16:50