Lebesgue integral of a nonnegative measurable function

Let \((X,\mcm,\mu)\) be a measure space and let \(f\colon X\to[0,\infty]\) be measurable. The Lebesgue integral of \(f\) is \[ \int f \coloneqq \sup \theset{\int \phi \suchthat \phi \text{ a simple function with } 0 \leq \phi \leq f}, \] where a nonnegative simple function \(\phi=\sum_{j=1}^n c_j\chi_{E_j}\) has integral \(\int\phi\coloneqq\sum_{j=1}^n c_j\,\mu(E_j)\).