Definition:Pre-Measure
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Definition
Let $X$ be a set.
Let $\SS \subseteq \powerset X$ be a collection of subsets of $X$.
Let $\O \in \SS$.
Let $\mu: \SS \to \overline \R$ be a mapping, where $\overline \R$ denotes the set of extended real numbers.
Then $\mu$ is said to be a pre-measure if and only if it satisfies the following conditions:
- $(1): \quad$ For all $A \in \SS$, if $\map \mu A$ is finite then $\map \mu A \ge 0$.
- $(2): \quad \map \mu \O = 0$
- $(3): \quad$ For every sequence $\sequence {A_n}_{n \mathop \in \N}$ of pairwise disjoint sets in $\SS$ with $\ds \bigcup_{n \mathop \in \N} A_n \in \SS$:
- $\ds \map \mu {\bigcup_{n \mathop \in \N} A_n} = \sum_{n \mathop \in \N} \map \mu {A_n}$
- that is, that $\mu$ is countably additive.
Also see
- Measure, a refinement which imposes that $\SS$ be a $\sigma$-algebra.
- Results about pre-measures can be found here.
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $4.1$