Lebesgue integral of a nonnegative function

Let \((X,\mcm,\mu)\) be a measure space. For a nonnegative simple function \(s = \sum_{j=1}^n c_j \chi_{E_j}\) with \(c_j\in[0,\infty)\) and \(E_j\in\mcm\), put \[ \int_X s \dmu \coloneqq \sum_{j=1}^n c_j \mu(E_j), \] with the convention \(0\cdot\infty=0\). For \(f\in\) $L^+$, the Lebesgue integral of \(f\) is \[ \int_X f \dmu \coloneqq \sup \theset{ \int_X s \dmu \suchthat s \text{ simple and } 0\leq s \leq f }. \]