Borel measure - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

Let $X$ be a topological space and $\mathcal B(X)$ its Borel sigma-algebra. A Borel measure is a measure $\mu:\mathcal B(X)\to[0,\infty]$ .

Additional hypotheses

“Borel measure” specifies the measurable sets, not regularity, finiteness, or local finiteness. Those are separate assumptions. A regular locally finite Borel measure on a locally compact Hausdorff space is often called a Radon measure.

Examples

Lebesgue measure restricted to Borel subsets of $\mathbb R^n$ and a Dirac measure $\delta_x$ , defined by $\delta_x(E)=1$ if $x\in E$ and $0$ otherwise, are Borel measures. Haar measure is a regular Borel measure with the additional translation-invariance property.

References

Gerald B. Folland, Real Analysis, 2nd ed., Wiley, 1999. Relevant: Chapter 7, Borel and regular measures.