ThmDex – An index of mathematical definitions, results, and conjectures.
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▼ Measurable space
Definition D1158
Measure space
Formulation 0
A D5107: Triple $M = (X, \mathcal{F}, \mu)$ is a measure space if and only if
(1) $(X, \mathcal{F})$ is a D1108: Measurable space
(2) $\mu$ is an D85: Unsigned basic measure on $(X, \mathcal{F})$
Subdefinitions
▶ Probability space
Children
▶ Almost everywhere constant map
▶ Measure-preserving endomorphism
▶ Probability space
▶ Pushforward measure
Results
▶ Binary partition additivity of unsigned basic measure
▶ Borel-Cantelli lemma
▶ Countable partition additivity of unsigned basic measure
▶ Countable subadditivity of measure
▶ Finite disjoint additivity of probability measure
▶ Finite disjoint additivity of unsigned basic measure
▶ Finite partition additivity of unsigned basic measure
▶ Inclusion-exclusion principle for probability of binary union
▶ Inclusion-exclusion principle for unsigned basic measure of binary union
▶ Inclusion-exclusion principle for unsigned basic measure of ternary union
▶ Isotonicity of unsigned basic measure
▶ Measure of finite union finite iff measure of all sets in union finite
▶ Measure of intersection finite if measure of at least one set finite
▶ Measure of measurable set complement
▶ Measure of set difference
▶ Measure of symmetric difference of set and subset
▶ Probability of symmetric difference of event and subevent
▶ Sequential continuity of measure from below