Measure, Measure Space, and Probability Measure

Let $(X,\mathcal{F})$ be a measurable space.

Write $[0,\infty]$ for the set $[0,\infty)\cup\{\infty\}$, where $\infty$ is a formal symbol with the conventions $a+\infty=\infty+a=\infty$ for all $a\in[0,\infty]$, $a<\infty$ for all real $a\ge 0$, and $0\cdot\infty=\infty\cdot 0=0$. The sum of a sequence $(a_m)_{m\in\mathbb{N}}$ in $[0,\infty]$ is defined as follows: if every $a_m$ is real and the partial sums are bounded above, then $\sum_m a_m$ is their least upper bound (which is also their limit, as the partial sums are nondecreasing); otherwise $\sum_m a_m=\infty$.

A measure on $(X,\mathcal{F})$ is a function $\mu:\mathcal{F}\to[0,\infty]$ such that $\mu(\varnothing)=0$ and, for every sequence $(A_m)_{m\in\mathbb{N}}$ of pairwise disjoint members of $\mathcal{F}$,

(countable additivity). The triple $(X,\mathcal{F},\mu)$ is a measure space.

The measure $\mu$ is finite if $\mu(X)<\infty$; it is $\sigma$-finite if there is a sequence $(X_m)_{m\in\mathbb{N}}$ in $\mathcal{F}$ with $X=\bigcup_m X_m$ and $\mu(X_m)<\infty$ for every $m$; and it is a probability measure if $\mu(X)=1$.