$v(B)=\int_{B} f d\mu $

Question

I have a question in integration theory:

If I have $(\Psi,\mathcal{G},\mu)$ a $\sigma$-finite measure space and $f$ a $[0,\infty]$-valued measurable function on $(\Psi,\mathcal{G})$ that is finite a.s.

So my question is if I define for $B\in \mathcal{G}$ $$v(B)=\int_{B} f d\mu $$

Is $(\Psi,\mathcal{G},v)$ a $\sigma$-finite measure space too ?

I think this reationship betwwen $v$ and $\mu$ can help me in calculational purpose.

Could someone help me? Thanks for the time and help.

Answer 1

If $\mu$ is $\sigma$-finite, there exists a countable collection of disjoint sets $X_i$ s.t. $\mu(X_i)<\infty$ and $\bigcup_{i\ge 1}X_i=X$. Consider $F_j=\{j-1\le f<j\}$, $j\in \mathbb{N}$. Then $\{X_i\cap F_j:i,j\in \mathbb{N}\}\cup\{f=\infty\}$ is a countable partition of $X$, s.t. each set in the partition has finite $v$ measure.