from finite to $\sigma$-finite measure space

Question

This might be rather elementary. I have put it at MSE for a while without getting any answers.

Here is the question: In the proof of the following theorem, would anyone explain how the general case (when $(\Omega,\Sigma,\mu)$ "$\sigma$-finite") can be deduced from the proved "finite" case?

Answer 1

The only point where the measures itself intervenes is in the definition of strong measurability, through simple convergence $\mu$-almost everywhere. A $\sigma$-finite measure is equivalent (in the sense has the same negligible sets) to a finite measure: partition $\Omega$ into countably many sets of finite $\mu$-measures, and define $d\nu=\rho\ d\mu$ where $\rho$ is positive, constant on each set of the partition, and decreases fast enough to make $\nu$ finite.

Answer 2

I've added an answer to the original question on MSE, which can be seen here: How can this theorem about weakly measurable functions on $\sigma$-finite measure spaces be deduced from the finite measure space case?. I have left it un-copied since it's not research-level.