Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space.
Let $f_1, \ldots, f_N : X \to [0, \infty]$ each be a D313: Measurable function on $M$.
Let $\lambda_1, \ldots, \lambda_N \in [0, \infty)$ each be an D4767: Unsigned real number.
Let $f_1, \ldots, f_N : X \to [0, \infty]$ each be a D313: Measurable function on $M$.
Let $\lambda_1, \ldots, \lambda_N \in [0, \infty)$ each be an D4767: Unsigned real number.
Then
\begin{equation}
\int_X \left( \sum_{n = 1}^N \lambda_n f_n \right) \, d \mu
= \sum_{n = 1}^N \lambda_n \left( \int_X f_n \, d \mu \right)
\end{equation}