Proof of A Probability Measure Is Finite

Proof of (i). If $\mu$ is a probability measure, then by Measure, Measure Space, and Probability Measure (v)

$$\mu(X)=1<\infty$$

by Real Numbers Lie Strictly below ∞, and Each Element of a Subset of [−∞,∞] Lies below Its Supremum (i) (as $1\in\mathbb{R}$), so $\mu$ is finite by Measure, Measure Space, and Probability Measure (iii).

Proof of (ii). By Probability Spaces, Events and Random Elements (i), $P$ is a probability measure on the measurable space $(\Omega,\mathcal{F})$, so $P$ is finite by (i). $\blacksquare$