Dirac Measure - Properties of The Dirac Measure

Properties of The Dirac Measure

Let δ_x denote the Dirac measure centred on some fixed point x in some measurable space (X, Σ).

• δ_x is a probability measure, and hence a finite measure.

Suppose that (X, T) is a topological space and that Σ is at least as fine as the Borel σ-algebra σ(T) on X.

• δ_x is a strictly positive measure if and only if the topology T is such that x lies within every non-empty open set, e.g. in the case of the trivial topology {∅, X}.
• Since δ_x is probability measure, it is also a locally finite measure.
• If X is a Hausdorff topological space with its Borel σ-algebra, then δ_x satisfies the condition to be an inner regular measure, since singleton sets such as {x} are always compact. Hence, δ_x is also a Radon measure.
• Assuming that the topology T is fine enough that {x} is closed, which is the case in most applications, the support of δ_x is {x}. (Otherwise, supp(δ_x) is the closure of {x} in (X, T).) Furthermore, δ_x is the only probability measure whose support is {x}.
• If X is n-dimensional Euclidean space Rn with its usual σ-algebra and n-dimensional Lebesgue measure λn, then δ_x is a singular measure with respect to λn: simply decompose Rn as A = Rn \ {x} and B = {x} and observe that δ_x(A) = λn(B) = 0.