I have been defining the Lebesgue integral with the lower Lebesgue sum. Can I use this to prove that if $(X, S, \mu)$ is a measure space with $\mu(X) < \infty$ and if $f : X → [0, \infty)$ is a bounded $S$ -measurable function. Then $\int f d \mu=\inf \left\{\sum_{j=1}^{m} \mu\left(A_{j}\right) \sup _{A_{j}} f: A_{1}, \ldots, A_{m} \text { is an } \mathcal{S} \text { -partition of } X\right\}$.
Any help would be much appreciated. Thanks.