The Measures, the Finite Measures and the Probability Measures on a Measurable Space Form Sets

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $\mathcal{F}$ be a $\sigma$-algebra on a set $X$, so that $(X,\mathcal{F})$ is a measurable space, let $[0,\infty]$ be as in The Extended Real Line, Its Order, and [0,∞] (ii), and let $[0,\infty]^{\mathcal{F}}$ be the set of functions from $\mathcal{F}$ to $[0,\infty]$ (The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set (ii)).

(i) Measures. There is a set $\mathcal{M}\subseteq[0,\infty]^{\mathcal{F}}$ such that, for every set $\mu$, $\mu\in\mathcal{M}$ if and only if $\mu$ is a measure on $(X,\mathcal{F})$.

(ii) Finite measures. There is a set $\mathcal{M}_{\mathrm{f}}\subseteq[0,\infty]^{\mathcal{F}}$ such that, for every set $\mu$, $\mu\in\mathcal{M}_{\mathrm{f}}$ if and only if $\mu$ is a finite measure on $(X,\mathcal{F})$.

(iii) Probability measures. There is a set $\mathcal{M}_{1}\subseteq[0,\infty]^{\mathcal{F}}$ such that, for every set $\mu$, $\mu\in\mathcal{M}_{1}$ if and only if $\mu$ is a probability measure on $(X,\mathcal{F})$.