Math 6210 • Real Analysis

Suppose that \((X,\mathfrak M,\mu )\) is a measure space with \(\mu \) a finite positive measure. Let \(I\) be an open interval in \(\mathbb R\) and \(f:X\times I\to \mathbb C\) a bounded function such that \(x\to f(x,t)\) is \(\mathfrak M\)-measurable for each \(t\in I\) and \(t\to f(x,t)\) is differentiable for each \(x\in X\). If \(\frac {d}{dt}f(x,t)\) is also bounded on \(X\times I\) prove that \(h(t)=\int _X f(x,t)\,d\mu (x)\) is differentiable on \(I\) and in fact \[ h'(t)=\int _X \frac {d}{dt}f(x,t)\,d\mu (x). \]