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 $M$.

Then \begin{equation} \int_X \left( \sum_{n = 1}^N f_n \right) \, d \mu = \sum_{n = 1}^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 $M$.

Then \begin{equation} \mu \left( \sum_{n = 1}^N f_n \right) = \sum_{n = 1}^N \mu(f_n) \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 $M$.

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