Integral with respect to Discrete Measure

Theorem

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.

Let $\ds \mu = \sum_{n \mathop \in \N} \lambda_n \delta_{x_n}$ be a discrete measure on $\struct {X, \Sigma}$.

Let $f \in \MM_{\overline \R}^+, f: X \to \overline \R$ be a positive measurable function.

Then:

$\ds \int f \rd \mu = \sum_{n \mathop \in \N} \lambda_n \map f {x_n}$

We have:

$\ds \int f \rd \mu$

$=$

$\ds \sum_{n \mathop \in \N} \lambda_n \int f \rd \delta_{x_n}$

$\ds $

$=$

$\ds \sum_{n \mathop \in \N} \lambda_n \map f {x_n}$

$\blacksquare$