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Assume that $\phi:\mathbb{R}\rightarrow\mathbb{R}$ is a strictly monotonically increasing function. Is it true that a real valued function $f:X\rightarrow\mathbb{R}$ is measurable on $(X,M)$ iff its composition $\phi \circ f$ with $\phi$ is measurable.

What if $\phi$ is just a monotone function?

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Hint for first part: $f$ is measurable if $f^{-1}([a,\infty)) \in M$ for all $a \in \mathbb R$.

Hint for second part: what if $\phi$ is constant?

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