Convergence in Distribution of Random Elements of a Metric Space

Let $(X,d)$ be a metric space. For each $n\in\mathbb{N}$ let $Y_n$ be a random element of $(X,d)$ on a probability space $(\Omega_n,\mathcal{F}_n,P_n)$, and let $Y$ be a random element of $(X,d)$ on a probability space $(\Omega,\mathcal{F},P)$; these probability spaces are not assumed to be the same. Let $\lambda_n$ denote the law of $Y_n$ and let $\lambda$ denote the law of $Y$.

We say that $(Y_n)_{n\in\mathbb{N}}$ converges in distribution to $Y$ if $(\lambda_n)_{n\in\mathbb{N}}$ converges weakly to $\lambda$.