PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-21

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A Dirac set function is a probability measure

Statement

For $x_{0} \in X$ , the Dirac set function $δ_{x_{0}}$ is a probability measure on every sigma-algebra on $X$ .

Facts & Assumptions

Given: A sigma-algebra $A$ on a nonempty set $X$ and a point $x_{0} \in X$ .

[L1]

The Dirac set function has value $1$ exactly on the measurable sets containing $x_{0}$ , and value $0$ otherwise (The Dirac set function at a point).

[L2]

A probability measure is a measure whose value on the whole space is $1$ (Probability measures and probability spaces).

[L3]

A nonnegative extended series is the supremum of its finite partial sums, with the empty sum equal to $0$ (Series in the nonnegative extended real line).

Proof

technique · direct

1.1givenL1

One has $δ_{x_{0}} (\emptyset) = 0$ and $δ_{x_{0}} (X) = 1$ .

1.2givenL1L3

If $(E_{k})$ is pairwise disjoint, then $x_{0}$ belongs to at most one $E_{k}$ . If it belongs to none, both $δ_{x_{0}} (⋃_{k} E_{k})$ and $\sum_{k} δ_{x_{0}} (E_{k})$ are $0$ ; if it belongs to the unique $E_{r}$ , both are $1$ .

2.1step 1.1step 1.2L2∎

Step 1.2 proves countable additivity and step 1.1 gives the empty-set and total-mass conditions, so $δ_{x_{0}}$ is a probability measure.

Depends on

Dependency tree · 2 levels

Sources

T. Tao, An Introduction to Measure Theory, Example 1.4.29