I have always had trouble to prove that something is well-defined I never know how to do it. I don't even know what do I have to prove. So I'm studying Lebesgue integration of non-negative functions and although it seems obvious for the author that $ \int f d \lambda $ is well defined, I wish I could understand perfectly why?
The formal definition is: Suppose $f: \mathbb{R}^n \rightarrow [0, \infty ]$ is measurable. Then the integral of $f$ is $$\int f d \lambda = \sup \left\{ \left. \int s d \lambda \: \right | \: s \leq f, s \in S \right\} $$
Where $\int s d \lambda = \sum_{k=1}^{m} \alpha_{k} \lambda (A_{k}).$
So I want to prove formally that the integral is well-defined. Thanks!