Math 2.0

Probability theory and stochastic processes Proposition

Continuity of probability measures

Builds on Probability spaceProbability measure

Statement

Let (Ω,ℱ,P)(\Omega, \mathcal{F}, P) be a probability space. If A1⊆A2⊆⋯A_1 \subseteq A_2 \subseteq \cdots are events and A=⋃n=1∞AnA = \bigcup_{n=1}^{\infty} A_n, then lim⁡n→∞P(An)=P(A)\lim_{n \to \infty} P(A_n) = P(A). If B1⊇B2⊇⋯B_1 \supseteq B_2 \supseteq \cdots are events and B=⋂n=1∞BnB = \bigcap_{n=1}^{\infty} B_n, then lim⁡n→∞P(Bn)=P(B)\lim_{n \to \infty} P(B_n) = P(B).

Remarks

The first part is continuity from below. Its proof writes AA as the disjoint union of A1A_1 and the sets An+1∖AnA_{n+1} \setminus A_n, and uses countable additivity. The second part, continuity from above, follows from the first part with the complements BncB_n^c. For a general measure μ\mu, continuity from above needs μ(B1)<∞\mu(B_1) < \infty. A probability measure is finite, so this holds. For example, P(X≤x+1/n)→P(X≤x)P(X \le x + 1/n) \to P(X \le x) for a real random variable XX. This gives the right continuity of the cumulative distribution function.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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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 measuredefinitionProbability space, definition, no evidence levelProbability spacedefinitionContinuity of probability measures, statement, no evidence levelContinuity of probabi…statement
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
  28. Probability space definition
  29. Continuity of probability measures statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Measure_(mathematics), Continuity from below
About this page
Tag0X8HBKCSYB
URIm2://0X8HBKCSYB
Aliasescontinuity-from-below, continuity-from-above
Kindstatement, proposition
WritingW0 stub
MSC60A10
StatusThis copy of the site has no build data. Run m2 status 0X8HBKCSYB.
UsesProbability space, Probability measure

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