Is it correct to say that if $E[z] < \infty$ then $z$ is almost surely finite?
2026-03-26 02:54:39.1774493679
Measure theory and almost surely
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In probability theory we consider only real valued measurable functions. If you are asking if $\int f \, d\mu <\infty$ implies $f$ is almost everywhere finite (in a general measure space setting) then the answer is no: take $f \equiv -\infty$. (You must always distinguish between $\int f \, d\mu <\infty$ and $\int f \, d\mu \in \mathbb R$)