A Probability Measure Is Finite

We work in the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation.

(i) Let $(X,\mathcal{F})$ be a measurable space and $\mu$ a measure on it. If $\mu$ is a probability measure, then $\mu$ is finite.

(ii) If $(\Omega,\mathcal{F},P)$ is a probability space, then $P$ is a measure on $(\Omega,\mathcal{F})$ that is finite.