A measurable function is the pointwise limit of simple functions with increasing moduli, and the convergence is uniform on every set where the function is bounded.Suggest a slogan
Let \((X,\mcm)\) be a measurable space and let \(f\colon X\to\CC\) be measurable. There is a sequence \((\phi_n)\) of simple functions such that \(0\leq\abs{\phi_1}\leq\abs{\phi_2}\leq\cdots\leq\abs{f}\), \(\phi_n\to f\) pointwise on \(X\), and \(\phi_n\to f\) uniformly on every set on which \(f\) is bounded [@Fol13].