Definition 16.1.2 (Integral of Non-Negative Simple Functions). Let $(X, \cm, \mu)$ be a measure space and $f = \sum_{y \in f(X)}y \cdot \one_{\bracs{f = y}}\in \Sigma^{+}(X, \cm)$ be a non-negative simple function in standard form, thenWith the convention that $0 \cdot \infty = 0$.
\[\int f d\mu = \int f(x) \mu(dx) = \sum_{y \in f(X)}y \cdot \mu(\bracs{f = y})\]
is the Lebesgue integral of $f$.