Math 2.0

Probability theory and stochastic processes Definition

Probability measure

Builds on Sigma-algebraProbability axiomsInterval

Definition

Let Ω\Omega be a set and let ℱ\mathcal{F} be a σ-algebra on Ω\Omega. A probability measure on (Ω,ℱ)(\Omega, \mathcal{F}) is a function P ⁣:ℱ→[0,1]P \colon \mathcal{F} \to [0, 1] with these two properties. P(Ω)=1P(\Omega) = 1. For each sequence A1,A2,…A_1, A_2, \dots of elements of ℱ\mathcal{F} with Ai∩Aj=∅A_i \cap A_j = \varnothing for all i≠ji \ne j, P(⋃n=1∞An)=∑n=1∞P(An).P\left(\bigcup_{n=1}^{\infty} A_n\right) = \sum_{n=1}^{\infty} P(A_n).

Remarks

These are the probability axioms. In other words, a probability measure is a measure of total mass 11. Countable additivity implies P(∅)=0P(\varnothing) = 0 and finite additivity: P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B) for disjoint AA and BB. On a finite set Ω\Omega with the power set as ℱ\mathcal{F}, each probability measure has the form P(A)=∑ω∈Ap(ω)P(A) = \sum_{\omega \in A} p(\omega) with p(ω)≥0p(\omega) \ge 0 and ∑ω∈Ωp(ω)=1\sum_{\omega \in \Omega} p(\omega) = 1.

Reading path

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Construction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptElement, definition, no evidence levelElementdefinitionIntegers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionProposition, concept, no evidence levelPropositionconceptRandom experiment, concept, no evidence levelRandom experimentconceptRational numbers, object, no evidence levelRational numbersobjectSet, definition, no evidence levelSetdefinitionRing of integers, statement, no evidence levelRing of integersstatementDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionUnion, definition, no evidence levelUniondefinitionSample space, definition, no evidence levelSample spacedefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionSubset, definition, no evidence levelSubsetdefinitionThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementPower set, definition, no evidence levelPower setdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionOrder on the reals, definition, no evidence levelOrder on the realsdefinitionComplement, definition, no evidence levelComplementdefinitionInterval, definition, no evidence levelIntervaldefinitionSigma-algebra, definition, no evidence levelSigma-algebradefinitionEvent, definition, no evidence levelEventdefinitionProbability axioms, concept, no evidence levelProbability axiomsconceptProbability measure, definition, no evidence levelProbability measuredefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Random experiment concept
  7. Rational numbers object
  8. Set definition
  9. Ring of integers statement
  10. Disjunction definition
  11. Implication definition
  12. Predicate definition
  13. Union definition
  14. Sample space definition
  15. The rationals form a field statement
  16. Universal quantifier definition
  17. Subset definition
  18. The rationals form an ordered field statement
  19. Power set definition
  20. Universal set definition
  21. Order on the reals definition
  22. Complement definition
  23. Interval definition
  24. Sigma-algebra definition
  25. Event definition
  26. Probability axioms concept
  27. Probability measure definition

Where this leads

373 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Probability_measure, Definition
About this page
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Kinddefinition
WritingW1 stated
MSC60A10
DefinesProbability measure
UsesSigma-algebra, Probability axioms, Interval

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