Random Element of a Metric Space and Its Law

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $(X,d)$ be a metric space with Borel $\sigma$-algebra $\mathcal{B}(X)$.

A random element of $(X,d)$ on $(\Omega,\mathcal{F},P)$ is a map $Y:\Omega\to X$ that is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(X)$.

The law of a random element $Y$ of $(X,d)$ is the image measure $P_Y$ of $P$ under $Y$, that is, the measure on $(X,\mathcal{B}(X))$ given by

$$P_Y(B)=P\bigl(Y^{-1}(B)\bigr)\qquad\text{for }B\in\mathcal{B}(X).$$