Math 2.0

Probability theory and stochastic processes Definition

Almost surely

Builds on Probability spaceEvent

Definition

Let (Ω,ℱ,P)(\Omega, \mathcal{F}, P) be a probability space. An event E∈ℱE \in \mathcal{F} happens almost surely if P(E)=1P(E) = 1. More generally, a property of outcomes holds almost surely (a.s.) if an event of probability 00 contains all outcomes where it fails.

Remarks

The property need not hold for all outcomes. In this sense, "almost surely" is weaker than "surely". Two random variables XX and YY are equal almost surely if P(X=Y)=1P(X = Y) = 1. They then have the same law. A countable intersection of events that happen almost surely also happens almost surely. It is the probability form of "almost everywhere" in measure theory.

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 spacedefinitionAlmost surely, definition, no evidence levelAlmost surelydefinition
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. Almost surely definition

Where this leads

13 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Almost_surely, Formal definition
About this page
Tag0R46A1XY60
URIm2://0R46A1XY60
Kinddefinition
WritingW1 stated
MSC60A05
DefinesAlmost surely
UsesProbability space, Event

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  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

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