# Definition: Measure

Let $$(M, {\mathcal {A}})$$ be a measurable set with the $$\sigma$$-algebra $$\mathcal{A}\,$$. A function mapping $$\mathcal {A}$$ to the set of positive real numbers $\mu \colon \cases{{\mathcal {A}}\longrightarrow {\mathbb {R}}_{+},\cr T\longmapsto \mu (T)},$ is called a measure on $$M$$, if $$\mu (\emptyset)=0$$ and if for every countable family of mutually disjoint subsets $$T_{i}\subseteq {\mathcal {A}}$$, $$i\in I$$, the measure of the union of these subsets equals the sum of the measures of each subset, formally: $\mu \left(\bigcup _{i\in I}T_{i}\right)=\sum _{i\in I}\mu (T_{i}).$ The second property is called $$\sigma$$-additivity.

Please note that the only difference between a measure and a pre-measure is that a measure is defined on a $$\sigma$$-algebra while a pre-measure is defined on a ring of sets.

Definitions: 1 2 3 4

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### References

#### Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of OsnabrÃ¼ck