Weak convergence of measures

Remarks

Common notations are $\mu_n \Rightarrow \mu$ and $\mu_n \xrightarrow{w} \mu$. Some authors call the same notion narrow convergence. Weak limits are unique on a metric space.

Example: let $\delta_a$ be the Dirac measure at $a$ on $\R$. Then $\delta_{1/n}$ converges weakly to $\delta_0$. Indeed $\int f \, d\delta_{1/n} = f(1/n)$, and $f(1/n) \to f(0)$ because $f$ is continuous.

Non-example: $\delta_n$ does not converge weakly to $\delta_0$. Take $f(x) = 1/(1 + x^2)$, which is bounded and continuous. Then $\int f \, d\delta_n = 1/(1 + n^2) \to 0$, but $\int f \, d\delta_0 = 1$.

Random variables converge in distribution exactly when their laws converge weakly.