Composition of monotonic and measurable function is measurable

Question

I am trying to find a counterexample since I think monotony is not enough.

Let $g:\mathbb{R} \to \mathbb{R}$ be monotone and for all $n \in \mathbb{N}, f: \mathbb{R}^n\to \mathbb{R}$ is measurable. Is $g \circ f$ and g measurable?

I know that a monotone function is measurable but I think for the composition $g$ must be continouous as well. What am I missing?

Answer 1

$g\circ f$ is measurable iff $(g\circ f)^{-1}\bigl((\alpha,\infty)\bigr)$ is measurable for al $\alpha$. By monotonicity of $g$, it is sufficient that $f$ is measurable.