ThmDex – An index of mathematical definitions, results, and conjectures.
▼ Set of symbols
▼ Alphabet
▼ Deduction system
▼ Theory
▼ Zermelo-Fraenkel set theory
▼ Set
▼ Subset
▼ Power set
▼ Hyperpower set sequence
▼ Hyperpower set
▼ Hypersubset
▼ Subset algebra
▼ Subset structure
▼ 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
▶ D1159: Probability space
Children
▶ D4248: Almost everywhere constant map
▶ D2940: Measure-preserving endomorphism
▶ D1159: Probability space
▶ D1732: Pushforward measure
Results
▶ R4927: Binary partition additivity of unsigned basic measure
▶ R1234: Borel-Cantelli lemma
▶ R3645: Countable partition additivity of unsigned basic measure
▶ R979: Countable subadditivity of measure
▶ R4841: Finite disjoint additivity of probability measure
▶ R976: Finite disjoint additivity of unsigned basic measure
▶ R4926: Finite partition additivity of unsigned basic measure
▶ R4445: Inclusion-exclusion principle for probability of binary union
▶ R4318: Inclusion-exclusion principle for unsigned basic measure of binary union
▶ R4317: Inclusion-exclusion principle for unsigned basic measure of ternary union
▶ R975: Isotonicity of unsigned basic measure
▶ R3970: Measure of finite union finite iff measure of all sets in union finite
▶ R3969: Measure of intersection finite if measure of at least one set finite
▶ R4277: Measure of measurable set complement
▶ R978: Measure of set difference
▶ R4443: Measure of symmetric difference of set and subset
▶ R4453: Probability of symmetric difference of event and subevent
▶ R982: Sequential continuity of measure from below