Finite measure

Remarks

Lebesgue measure on the Borel sets of $[0, 1]$ is finite, since $\mu([0, 1]) = 1$. Lebesgue measure on the Borel sets of $\R$ is not finite, since $\mu(\R) = \infty$. If $\mu$ is finite, then $\mu(A) \le \mu(X) < \infty$ for every $A \in \Sigma$, by monotonicity. Every probability measure is finite. If $0 < \mu(X) < \infty$, then $\mu / \mu(X)$ is a probability measure.