Math 2.0

Measure and integration Definition

Dirac measure

Builds on Sigma-algebraProbability measureIndicator function

Definition

Let XX be a set, let ℱ\mathcal{F} be a σ-algebra on XX, and let x∈Xx \in X. The Dirac measure at xx is the function δx ⁣:ℱ→{0,1}\delta_x \colon \mathcal{F} \to \{0, 1\} with δx(A)={1if x∈A,0if x∉A.\delta_x(A) = \begin{cases} 1 & \text{if } x \in A, \\ 0 & \text{if } x \notin A. \end{cases}

Remarks

δx\delta_x is a probability measure. In terms of the indicator function, δx(A)=𝟏A(x)\delta_x(A) = \mathbf{1}_A(x). The integral of a measurable function ff with respect to δx\delta_x is f(x)f(x). Each probability measure on a countable set XX with the power set as σ-algebra is a sum ∑x∈Xp(x) δx\sum_{x \in X} p(x) \, \delta_x with weights p(x)≥0p(x) \ge 0 of sum 11.

Reading path

To understand this page from the ground up, read these 35 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 levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionUnion, definition, no evidence levelUniondefinitionSample space, definition, no evidence levelSample spacedefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionOrder on the reals, definition, no evidence levelOrder on the realsdefinitionPower set, definition, no evidence levelPower setdefinitionComplement, definition, no evidence levelComplementdefinitionFunction, definition, no evidence levelFunctiondefinitionInterval, definition, no evidence levelIntervaldefinitionIndicator function, definition, no evidence levelIndicator functiondefinitionSigma-algebra, definition, no evidence levelSigma-algebradefinitionEvent, definition, no evidence levelEventdefinitionProbability axioms, concept, no evidence levelProbability axiomsconceptProbability measure, definition, no evidence levelProbability measuredefinitionDirac measure, definition, no evidence levelDirac 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.
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  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. Ordered pair definition
  14. Union definition
  15. Sample space definition
  16. The rationals form a field statement
  17. Existential quantifier definition
  18. Universal quantifier definition
  19. Cartesian product definition
  20. Subset definition
  21. The rationals form an ordered field statement
  22. Uniqueness quantifier definition
  23. Binary relation definition
  24. Universal set definition
  25. Order on the reals definition
  26. Power set definition
  27. Complement definition
  28. Function definition
  29. Interval definition
  30. Indicator function definition
  31. Sigma-algebra definition
  32. Event definition
  33. Probability axioms concept
  34. Probability measure definition
  35. Dirac measure definition

Where this leads

48 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Dirac_measure, Definition
About this page
Tag099SXAG014
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Kinddefinition
WritingW1 stated
MSC28A12
DefinesDirac measure
UsesSigma-algebra, Probability measure, Indicator function

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