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:

\(\ds \int_D f \rd \mu\)

\(=\)

\(\ds \int \paren {f \times \chi_D} \rd \mu\)

Definition of Integral of Positive Measurable Function over Measurable Set

\(\ds \)

\(=\)

\(\ds \int \paren {f \times \paren {\sum_{n \mathop = 1}^\infty \chi_{D_n} } } \rd \mu\)

Characteristic Function of Disjoint Union

\(\ds \)

\(=\)

\(\ds \int \paren {\sum_{n \mathop = 1}^\infty \paren {f \times \chi_{D_n} } } \rd \mu\)

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 1}^\infty \paren {\int \paren {f \times \chi_{D_n} } \rd \mu}\)

Integral of Series of Positive Measurable Functions

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 1}^\infty \int_{D_n} f_n \rd \mu\)

Definition of Integral of Positive Measurable Function over Measurable Set

$\blacksquare$