How to prove $\phi'(t)1_{\Omega_t}(w)$ is measurable?

Question

I need help or any hint in the next exercise:

Let $(\Omega,\mathcal{F},\mu)$ be a $\sigma-$finite measurable space and let $f:\Omega\to \mathbb{R}$ be a measurable function.

Let $\phi:\mathbb{R}^+\to\mathbb{R}^+ $ be a increasing function and differentiable, such that $\phi(0)=0$. for all $t>0$ it defines $\Omega_t=\{\omega \in \Omega:|f(w)|>t\}$.

Prove $(t,\omega)\to\phi'(t)1_{\Omega_t}(\omega)$ is positive and $\mathcal{B}(\mathbb{R}^+)\otimes\mathcal{F}-$measurable.

I have to start by seeing if $\phi '$ is measurable?

Answer 1

Hint:

$(\omega,t)\mapsto\phi'(t)$ is the pointwise limit of $G_n:(\omega,t)\mapsto n\big(\phi(x+\tfrac{1}{n})-\phi(x)\Big)$ and so it is measurable in the product..

The function $(\omega,t)\mapsto |f(\omega)|-t$ is measurable in the product, and so $E=\{(\omega,t):|f(\omega)|-t>0\}$ is measurable. Hence, the cross section $E_t=\{\omega:|f(\omega)|>t\}$ is measurable.

product of measurable functions is measurable (start with simple functions and then by approximation by simple functions).