Integral of Positive Simple Function is Well-Defined

Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $f: X \to \R, f \in \mathcal{E}^+$ be a positive simple function.

Then the $\mu$-integral of $f$, $I_\mu \left({f}\right)$, is well-defined.

That is, for any two standard representations for $f$, say:


 * $\displaystyle f = \sum_{i = 0}^n a_i \chi_{E_i} = \sum_{j = 0}^m b_j \chi_{F_j}$

it holds that:


 * $\displaystyle \sum_{i = 0}^n a_i \mu \left({E_i}\right) = \sum_{j = 0}^m b_j \mu \left({F_j}\right)$