An integrable function can be infinite only on a null set.Suggest a slogan
Let \((X,\mcm,\mu)\) be a measure space and let \(f\colon X\to[-\infty,\infty]\) be measurable with \(\int_X\abs{f}\dmu<\infty\). Then \[ \mu\qty{\theset{x\in X\suchthat \abs{f(x)} = \infty}} = 0 . \]
Let \(A\coloneqq\theset{x\in X\suchthat \abs{f(x)}=\infty}\). For every \(t>0\), \(\abs{f}\geq t\,\one_A\), so \(t\,\mu(A)\le\int_X\abs{f}\dmu\). Letting \(t\to\infty\) gives \(\mu(A)=0\).