TITLE: Integration respect to two measures
QUESTION [1 upvotes]: Let $(X,\mathcal{K},\mu_1)$ and $(X,\mathcal{K},\mu_2)$ be two  measure spaces, and let $\mu=\mu_1+\mu_2$. Assume $f:X\to[0,+\infty]$ are integrable both respect to $\mu_1$ and $\mu_2$, how do we prove that $f$ is also integrable respect to $\mu$ ?

REPLY [1 votes]: Hint
You have to prove that
$$\int f\mathrm d \mu=\int f\mathrm d \mu_1+\int f\mathrm d \mu_2<\infty.$$
It's not obvious, but not difficult to prove. Consider first $f=\boldsymbol 1_A$ where $A$ is $\mu_1$ and $\mu_2$ measurable. Then do the cases where $f$ is simple, positive and $\mu_1$,$\mu_2$ measurable, and finally $\mu_1$ and $\mu_2$ integrable. Then the claim will follow.
Notice that $f$ is $\mu_1$ and $\mu_2$ measurable $\implies $ $f$ is $\mu-$measurable (even if it's "natural", it might be good to prove if anyway). The converse is wrong.
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