For an almost everywhere increasing sequence in $L^+$, the integral of the limit is the limit of the integrals.Suggest a slogan
Let \((X,\mcm,\mu)\) be a measure space, and let \(f_n\) for \(n\geq1\) and \(f\) be functions in $L^+(X,\mcm)$ such that, for almost every \(x\in X\), \(f_n(x)\leq f_{n+1}(x)\) for all \(n\) and \(f_n(x)\to f(x)\). Then \[ \lim_{n\to\infty} \int_X f_n\dmu = \int_X f\dmu . \]