In mathematics, a Dirac measure is a measure δx on a set X that gives a given element x measure 1, so that
δx({x}) = 1
and in general
δx(Y) = 0
for any subset Y of X not containing x,
δx(Z) = 1
for any subset Z containing x.
The Dirac measure is a probability measure, and in terms of probability it represents the almost sure outcome x in the sample spaceX. We can also say that the measure is a single atom at x. The Dirac measures are the extreme points of the convex set of probability measures on X.
Measure informs analysis and is one of the key building blocks of the modern theory of analysis and probability.
Measure Property 2: The value of μ under any finite or countably infinite disjoint union of subsets of X that are also elements of M is equal to the finite or countably infinite sum respectively of the value of μ under each of the subsets.
Measure allows the expansion of the definition of the integral to functions whose domain is any arbitrary set with a corresponding σ-algebra and measure.
Dirac was born in Bristol, England on August 8, 1902 to Charles Adrien Ladislas Dirac, a Swiss immigrant, and Florence Hannah (Holten) Dirac, a native of Britain.
Dirac married Margit Wigner of Budapest in 1937.
Dirac was the Lucasian Professor of Mathematics at Cambridge from 1932 to 1969.
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