I've been reading the chapter of Uniform Integrability on Probability Theory by Achim Klenke which has the following proposition.


**If $\mu$ is $\sigma$-finite, then there is a function $h \in L^{1}(\mu)$ such that $h > 0$ almost everywhere.**   

In the proof, he constructed an increasing sequence of sets $A\_{n}$ with limit being the whole space $\Omega$ and each element possessing finite measure, i.e. $\mu(A\_{n}) < \infty$.


I am not sure whether this is only for $\mu$ being $\sigma$-finite, since in the following theorem he showed result similar to the Vitali convergence theorem but without the condition that restrict set measure.



On conditions for the existence of $h\in L^1$ such that $h>0$ a.e
It is false without assuming that the measure is $\sigma$-finite. Let $\mu$ be the counting measure on $\mathbb{R}$. Then $h>0$ a.e. means $h>0$ everywhere and clearly
$$
\int\_{\mathbb{R}}hd\mu=\sum\_{x\in\mathbb{R}} h(x)=+\infty.
$$


