How to show that let $(X,\mathcal{M},\mu)$ be a measurable space, there exists a function $f \in L^1(X,\mu)$ with $f>0$ $\mu$-a.e. iff $\mu$ is $\sigma$-finite. Can you please help me out? Thank you.
How to prove here exists a function $f \in L^1(X,\mu)$ with $f>0$ $\mu$-a.e. iff $\mu$ is $\sigma$-finite.
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real-analysis
functional-analysis