How to prove these sets are in the $\sigma$-algebra?

Question

Given a measure space $(\Omega,\mathcal M, \mu)$, I want to show $\nu(E)=\int_Efd\mu$ is $\sigma$-finite, where $f$ is nonnegative and $f\in L^1(\mu)$. The definition of $\sigma$-finite measure I use is that if $\Omega = \cup_1^\infty E_n$ where $E_n \in \mathcal M$ and $\nu(E_n) < \infty$ for all $n$, then $\nu$ is a $\sigma$-finite measure. So, I write $$\Omega=\bigcup_{n=1}^\infty \left\{x\in \Omega:|f(x)|\geq \frac{1}{n}\right\}\cup \{{ x\in \Omega: f(x)=0\}}.$$ But, how to show that$\left\{x\in \Omega:|f(x)|\geq \frac{1}{n}\right\}$ and $\{{ x\in \Omega: f(x)=0\}}$ are in $\mathcal M$?

Answer 1

But $f\in L^{1}(\mu)=L^{1}(\Omega,\mu)$ guarantees that $\nu(\Omega)=\displaystyle\int_{\Omega}fd\mu<\infty$, so $\nu$ is a finite measure, no?