Borel Sets of a Variable Metric Space as a Predicative Condition, and the Families of σ-Algebras Generated by Random Elements and of Their Composites

In the setting of Probability Spaces: Notation; the first five clauses are statements of the metatheory. Conditions expressed by a predicative formula are as in Conditions Expressed by a Predicative Formula, and Guards (ii), with parameters and their fixed values as in Conditions Expressed by a Predicative Formula, and Guards (i); the parameters $\mathbb{R}$, $\le$, $<$, $+$, $0$ and $\mathbb{N}$ have the fixed values the set of real numbers, its order, its strict order, its addition, its zero, and the set $\mathbb{N}$ of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion (ii). Families indexed by a set $J$ are as in Indexed Families of Sets and Their Union, Intersection and Product (i).

(i) Metric spaces. Let $S$ and $d$ be distinct set variables. The condition "$(S,d)$ is a metric space" is expressed by a predicative formula with parameters $\mathbb{R}$, $\le$, $+$ and $0$.

(ii) Open balls. Let $S$, $d$, $x$, $r$ and $y$ be distinct set variables. The condition "$(S,d)$ is a metric space, $x\in S$, $r\in\mathbb{R}$, $r>0$ and $y\in B_{d}(x,r)$", with the open ball $B_{d}(x,r)$, is expressed by a predicative formula with parameters $\mathbb{R}$, $\le$, $<$, $+$ and $0$.

(iii) Open sets. Let $S$, $d$ and $U$ be distinct set variables. The condition "$(S,d)$ is a metric space and $U\in\mathcal{T}_{d}$", with $\mathcal{T}_{d}$ the set of subsets of $S$ that are open in $(S,d)$, is expressed by a predicative formula with parameters $\mathbb{R}$, $\le$, $<$, $+$ and $0$.

(iv) $\sigma$-algebras on a variable set. Let $G$ and $Y$ be distinct set variables. The condition "$G$ is a $\sigma$-algebra on $Y$" is expressed by a predicative formula with parameter $\mathbb{N}$.

(v) Borel sets of a variable metric space. Let $B$, $S$ and $d$ be distinct set variables. The condition "$(S,d)$ is a metric space and $B\in\mathcal{B}(S,d)$", with the Borel $\sigma$-algebra $\mathcal{B}(S,d)$, is expressed by a predicative formula with parameters $\mathbb{R}$, $\le$, $<$, $+$, $0$ and $\mathbb{N}$.

(vi) The family $(\sigma(X_{j}))_{j\in J}$. Let $J$ be a set and let $(S_{j})_{j\in J}$, $(d_{j})_{j\in J}$ and $(X_{j})_{j\in J}$ be families such that, for every $j\in J$, $(S_{j},d_{j})$ is a metric space and $X_{j}$ is a random element of $S_{j}$. Then there is exactly one family $(\mathcal{G}_{j})_{j\in J}$ with

$$\mathcal{G}_{j}=\sigma(X_{j})\qquad(j\in J),$$

where $\sigma(X_{j})$ is the $\sigma$-algebra generated by $X_{j}$.

(vii) The family $(g_{j}(X_{j}))_{j\in J}$. Let $J$ be a set and let $(S_{j})_{j\in J}$, $(T_{j})_{j\in J}$, $(X_{j})_{j\in J}$ and $(g_{j})_{j\in J}$ be families such that, for every $j\in J$, $X_{j}:\Omega\to S_{j}$ and $g_{j}:S_{j}\to T_{j}$ are maps. Then there is exactly one family $(Y_{j})_{j\in J}$ with

$$Y_{j}=g_{j}(X_{j})\qquad(j\in J).$$