Math 6210 • Real Analysis

Let \((X,M,\mu )\) be a measure space, and suppose that \(f:X\times [0,1]\to \mathbb R\) such that \(f(\cdot ,t):X\to \mathbb R\) is integrable for all \(t\). Define \[ F(t)=\int _X f(x,t)\,d\mu (x) \] for \(t\in [0,1]\). Suppose that \(\frac {\partial f}{\partial t}(x,t)\) exists for all \(x\) and \(t\), and that for some integrable function \(g\), \(\left |\frac {\partial f}{\partial t}(x,t)\right |\le g(x)\) for all \(x\) and \(t\). Show that \(F\) is differentiable and that \[ F'(t)=\int _X \frac {\partial f}{\partial t}(x,t)\,d\mu (x) \]