Dirac measure

Let X be a nonempty set. Let 𝒫⁢(X) denote the power setMathworldPlanetmath of X. Then (X,𝒫⁢(X)) is a measurable spaceMathworldPlanetmathPlanetmath.

Let x∈X. The Dirac measure concentrated at x is δx:𝒫⁢(X)→{0,1} defined by

δx⁢(E)={1if ⁢x∈E0if ⁢x∉E.

Note that the Dirac measure δx is indeed a measureMathworldPlanetmath:

  1. 1.

    Since x∉∅, we have δx⁢(∅)=0.

  2. 2.

    If {An}n∈ℕ is a sequence of pairwise disjoint subsets of X, then one of the following must happen:

    • –

      x∉⋃n∈ℕAn, in which case δx⁢(⋃n∈ℕAn)=0 and δx⁢(An)=0 for every n∈ℕ;

    • –

      x∈⋃n∈ℕAn, in which case x∈An0 for exactly one n0∈ℕ, causing δx⁢(⋃n∈ℕAn)=1, δx⁢(An0)=1, and δx⁢(An)=0 for every n∈ℕ with n≠n0.

Also note that (X,𝒫⁢(X),δx) is a probability space.

Let ℝ¯ denote the extended real numbers. Then for any function f:X→ℝ¯, the integral of f with respect to the Dirac measure δx is


In other words, integration with respect to the Dirac measure δx amounts to evaluating the function at x.

If X=ℝ, m denotes Lebesgue measureMathworldPlanetmath, A is a Lebesgue measurable subset of ℝ, and δ (no ) denotes the Dirac delta function, then for any measurable functionMathworldPlanetmath f:ℝ→ℝ, we have


Moreover, if f is defined so that f⁢(t)=1 for all t∈A, the above becomes


In other words, the function δ⁢(t-x) (with x∈ℝ fixed and t a real variable) behaves like a Radon-Nikodym derivativeMathworldPlanetmath of δx with respect to m.

Note that, just as the Dirac delta function is a misnomer (it is not really a function), there is not really a Radon-Nikodym derivative of δx with respect to m, since δx is not absolutely continuousMathworldPlanetmath with respect to m.

Title Dirac measure
Canonical name DiracMeasure
Date of creation 2013-03-22 17:19:40
Last modified on 2013-03-22 17:19:40
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 18
Author Wkbj79 (1863)
Entry type Definition
Classification msc 60A10
Classification msc 26A42
Classification msc 28A25
Classification msc 28A12
Classification msc 28A10
Related topic Measure
Related topic Integral2
Related topic DiracDeltaFunction