Dirac measure
Let X be a nonempty set. Let 𝒫(X) denote the power set of X. Then (X,𝒫(X)) is a measurable space
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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 measure:
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1.
Since x∉∅, we have δx(∅)=0.
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2.
If {An}n∈ℕ is a sequence of pairwise disjoint subsets of X, then one of the following must happen:
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x∉⋃n∈ℕAn, in which case δx(⋃n∈ℕAn)=0 and δx(An)=0 for every n∈ℕ;
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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.
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Also note that (X,𝒫(X),δx) is a probability space.
Let ˉℝ denote the extended real numbers. Then for any function f:X→ˉℝ, the integral of f with respect to the Dirac measure δx is
∫Xf𝑑δx=f(x). |
In other words, integration with respect to the Dirac measure δx amounts to evaluating the function at x.
If X=ℝ, m denotes Lebesgue measure, A is a Lebesgue measurable subset of ℝ, and δ (no ) denotes the Dirac delta function, then for any measurable function
f:ℝ→ℝ, we have
∫Aδ(t-x)f(t)𝑑m(t)=∫Af𝑑δx=f(x)δx(A). |
Moreover, if f is defined so that f(t)=1 for all t∈A, the above becomes
∫Aδ(t-x)𝑑m(t)=∫A𝑑δx=δx(A). |
In other words, the function δ(t-x) (with x∈ℝ fixed and t a real variable) behaves like a Radon-Nikodym derivative 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 continuous 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 |