The Finite Signed Measures on a Measurable Space Form a Set

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, and let $\mathbb{R}^{\mathcal{F}}$ be the set of functions from $\mathcal{F}$ to $\mathbb{R}$ (The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set (ii)).

(i) Finite signed measures. There is a set $\mathcal{M}\subseteq\mathbb{R}^{\mathcal{F}}$ such that, for every set $\nu$, $\nu\in\mathcal{M}$ if and only if $\nu$ is a finite signed measure on $(X,\mathcal{F})$.