Proof of The Distance of Two Random Elements of a Separable Metric Space Is a Random Variable

Throughout, $d$ is nonnegative and symmetric and satisfies the triangle inequality (Metric Space (ii), Metric Space (iv), Metric Space (v)), and countable unions and intersections of events are events (Unions and Intersections over a Countable Index Set in a σ-Algebra (i), Unions and Intersections over a Countable Index Set in a σ-Algebra (ii)). The formula $\omega\mapsto d(X(\omega),Y(\omega))$ defines a function $\Omega\to\mathbb{R}$ by Maps and Relations Given by Formulas (i) and Maps and Relations Given by Formulas (ii); we write $d(q,X)$ for $\omega\mapsto d(q,X(\omega))$, and similarly for $Y$.

Step 1 (balls give events). For $q\in S$ and real $\rho>0$, the open ball $B_{d}(q,\rho)$ is open (Open Ball in a Metric Space is Open (i)), hence belongs to the $\sigma$-algebra generated by the open sets, which is $\mathcal{B}(S)$ (Borel Sigma-Algebra of a Metric Space (i), The σ-Algebra Generated by a Set of Subsets (i)). As $X$ and $Y$ are random elements of $S$ (Probability Spaces, Events and Random Elements (iii), Measurable Map, and the Set of Measurable Maps (i)), the sets

$$\{d(q,X)<\rho\}=X^{-1}\bigl(B_{d}(q,\rho)\bigr),\qquad\{d(q,Y)<\rho\}=Y^{-1}\bigl(B_{d}(q,\rho)\bigr)$$

are events.

Step 2 (the index set and its events). Let $D\subseteq S$ be a countable dense set (Separable Metric Space and Separable Real Normed Space (i)), and let $\mathbb{Q}_{\mathbb{R}}=\kappa[\mathbb{Q}]\subseteq\mathbb{R}$ be the image of the rationals under the canonical embedding $\kappa$. Fix a real number $t$. For $(q,r)\in D\times\mathbb{Q}_{\mathbb{R}}$ let

$$E(q,r)=\{\omega\in\Omega:\ d(q,X(\omega))<r,\ d(q,Y(\omega))+r<t\}.$$

Each $E(q,r)$ is an event. If $0<r<t$, then

$$E(q,r)=\{d(q,X)<r\}\cap\{d(q,Y)<t-r\},$$

an intersection of two events of Step 1. If $r\le0$, then $E(q,r)=\emptyset$, since $d(q,X(\omega))\ge0$; and if $r\ge t$, then also $E(q,r)=\emptyset$, since $d(q,Y(\omega))+r\ge r\ge t$.

Step 3 (the sets below a level). We show

$$\{d(X,Y)<t\}=\bigcup_{(q,r)\in D\times\mathbb{Q}_{\mathbb{R}}}E(q,r).$$

For $\supseteq$, let $\omega\in E(q,r)$. By the triangle inequality and symmetry,

$$d\bigl(X(\omega),Y(\omega)\bigr)\le d\bigl(q,X(\omega)\bigr)+d\bigl(q,Y(\omega)\bigr)<r+d\bigl(q,Y(\omega)\bigr)<t .$$

For $\subseteq$, let $\delta=d(X(\omega),Y(\omega))<t$. By the density of the rationals (The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (x), Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema (iii)) there is $c\in\mathbb{Q}_{\mathbb{R}}$ with $\delta<c<t$. Let $\eta=\min(c-\delta,\,t-c)>0$, so that

$$\delta+2\eta\le\delta+(c-\delta)+(t-c)=t .$$

Density of $D$ gives $q\in D$ with $d(q,X(\omega))<\eta$, and density of the rationals gives $r\in\mathbb{Q}_{\mathbb{R}}$ with $d(q,X(\omega))<r<\eta$. Then, by the triangle inequality,

$$d\bigl(q,Y(\omega)\bigr)+r\le d\bigl(q,X(\omega)\bigr)+\delta+r<\eta+\delta+\eta\le t ,$$

so $\omega\in E(q,r)$.

Step 4 (a countable union of events). The set $\mathbb{Q}_{\mathbb{R}}$ is countable as the image of the countable set $\mathbb{Q}$ (Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (ix), Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (vi)), so $D\times\mathbb{Q}_{\mathbb{R}}$ is countable by Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (vii). By Steps 2 and 3, $\{d(X,Y)<t\}$ is a countable union of events, hence an event.

Step 5 (conclusion). As $t$ was arbitrary, criterion (d) of Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (v) shows that $d(X,Y)$ is measurable as a function into $\mathbb{R}$, that is, a random variable (Probability Spaces, Events and Random Elements (iv)).