Integral of a non-negative function

Remarks

Example: let $X = \N$ with the counting measure and $f(n) = 2^{-n}$ for $n \ge 1$, $f(0) = 1$. The simple functions $s_N = f \cdot \mathbf{1}_{\set{0, \dots, N}}$ lie below $f$, and their integrals $\sum_{n=0}^{N} 2^{-n}$ increase to $2$. Every simple $s \le f$ has integral at most $\sum_n f(n) = 2$, so $\int f \, d\mu = 2$.

Non-example: the formula does not apply to $g(x) = \sin x$ on $\R$, because $g$ takes negative values. Such functions need the splitting $g = g^+ - g^-$ of the Lebesgue integral.

The value of $\int_X s \, d\mu$ for a simple function does not depend on the way $s$ is written. For $A \in \Sigma$ one writes $\int_A f \, d\mu = \int_X f \mathbf{1}_A \, d\mu$. If $f = g$ almost everywhere, then $\int_X f \, d\mu = \int_X g \, d\mu$.