ThmDex – An index of mathematical definitions, results, and conjectures.
Formulation F10519 on R3586: Basic function is finite almost everywhere iff positive and negative parts are
F10519
Formulation 2
Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that
(i) $f : X \to [-\infty, \infty]$ is a D5600: Basic Borel function on $M$
Then $|f| \overset{a.e.}{<} \infty$ if and only if
(1) \begin{equation} f^+ \overset{a.e.}{<} \infty \end{equation}
(2) \begin{equation} f^- \overset{a.e.}{<} \infty \end{equation}