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.

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}

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.

Then \begin{equation} \int_X \left(\lambda_1 f_1 + \cdots + \lambda_N f_N \right) \, d \mu = \lambda_1 \int_X f_1 \, d \mu + \cdots + \lambda_N \int_X f_N \, d \mu \end{equation}