Math 6210 • Real Analysis

Let \((X,\mu )\) be a measure space, let \(\{f_n\}\) be a sequence of nonnegative integrable functions, and assume there exists an integrable function \(f\) such that \(f(x)=\lim _{n\to \infty } f_n(x)\) almost everywhere. Further assume \[ \tag {1} \lim _{n\to \infty }\int _X f_n\,d\mu =\int _X f\,d\mu . \] Prove that \[ \lim _{n\to \infty }\int _X |f_n-f|\,d\mu =0. \] Show by example that this need not hold if the assumption (1) is omitted.