<Weak convergence of probability measures> $\mu_n\to\mu$ on a <metric space> means that
$$
\int f\,d\mu_n\longrightarrow\int f\,d\mu
$$
for every bounded continuous real function $f$. <Weak convergence of random variables> means weak convergence of their <probability distributions>; equivalently,
$$
\mathbb E[f(X_n)]\longrightarrow\mathbb E[f(X)]
$$
for every bounded continuous $f$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 2 / a / Solution

Probability measures $\mu_n$ on a metric space converge weakly to $\mu$ when $\int f\,d\mu_n\to\int f\,d\mu$ for every bounded continuous real function $f$.


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