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 $f:Omegato mathbb{R}$ be a measurable function.
Let $phi:mathbb{R}^+tomathbb{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 $(t,omega)tophi'(t)1_{Omega_t}(omega)$ is positive and $mathcal{B}(mathbb{R}^+)otimesmathcal{F}-$measurable.
I have to start by seeing if $phi '$ is measurable?

Oliver Diaz · Answer

Hint:
$(omega,t)mapstophi'(t)$ is the pointwise limit of $G_n:(omega,t)mapsto nbig(phi(x+tfrac{1}{n})-phi(x)Big)$ 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).

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

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