Integral with respect to Discrete Measure

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Theorem

Let $\left({X, \Sigma}\right)$ be a measurable space.

Let $\displaystyle \mu = \sum_{n \mathop \in \N} \lambda_n \delta_{x_n}$ be a discrete measure on $\left({X, \Sigma}\right)$.


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


Then:

$\displaystyle \int f \, \mathrm d\mu = \sum_{n \mathop \in \N} \lambda_n f \left({x_n}\right)$

where the integral sign denotes $\mu$-integration.


Proof


Sources