Integral of Positive Measurable Function over Disjoint Union

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $f : X \to \overline \R$ be a positive $\Sigma$-measurable function.

Let $\sequence {D_n}_{n \mathop \in \N}$ be a sequence of pairwise disjoint $\Sigma$-measurable sets.

Let:


 * $\ds D = \bigcup_{n \mathop = 1}^\infty D_n$

Then:


 * $\ds \int_D f \rd \mu = \sum_{n \mathop = 1}^\infty \int_{D_n} f \rd \mu$

Proof
We have: