Borel measure

Remarks

Example: the Dirac measure $\delta_p$ at a point $p \in X$, with $\delta_p(A) = 1$ if $p \in A$ and $0$ otherwise, is a Borel measure. It is defined on all subsets of $X$, hence on $\mathcal{B}(X)$, and it is countably additive because $p$ lies in at most one set of a disjoint family.

Non-example: on $\R$, let $\Sigma = \set{A \subseteq \R : A \text{ or } \R \setminus A \text{ is countable}}$ and let $\mu$ be $0$ on countable sets and $1$ on the others. Then $\mu$ is a measure on $\Sigma$, but $\Sigma$ does not contain the interval $(0, 1)$, so $\mu$ is not defined on all Borel sets and is not a Borel measure.

Conventions vary. Some authors call a measure on a σ-algebra containing $\mathcal{B}(X)$, such as Lebesgue measure on the Lebesgue sets, a Borel measure. Some authors require in addition that $\mu(K) < \infty$ for every compact set $K$; then counting measure on $\R$ is not a Borel measure.