Definition 22.2.2 (Integral of Non-Negative Function).label Let $(X, \cm, \mu)$ be a measure space and $f \in \mathcal{L}^{+}(X, \cm)$, then
\[\int f d\mu = \int f(x)\mu(dx) = \sup\bracs{\int \phi d\mu \bigg | \phi \in \Sigma^+(X, \cm), \phi \le f}\]
is the Lebesgue integral of $f$.