Let $(X, \mathcal A, \mu)$ be a measure space. I need to show that $\int _X f \:\:d\mu$ is NOT dependent on the representation of the simple function $f$. That means: For $M, N \in \Bbb N ; \: \: \alpha_1 , \dots , \alpha _N \lt 0; \: \:\beta_1, \dots \beta _ M \lt 0 ;\: \: A_1 , \dots , A_N \in \mathcal A$ and $B_1 , \dots , B_M \in \mathcal A$ are chosen so that: $$f=\sum_{i=1}^{N} \alpha _i \chi_{A_i}=\sum_{i=1}^{M} \beta _i \chi_{B_i} $$ it is true that: $$\sum_{i=1}^{N} \alpha _i \mu (A_i)=\sum_{i=1}^{M} \beta _i \mu (B_i) $$
It's obvious that this should be the case, but how do i write it down in a formal prove?
Any tipps or ideas? Thanks in advance!