# Convergence in probability implies weakly convergence of sequence of induced measures

I was stuck the following problem: Let $X_n$, $X$, $n\ge 1$, be random variables defined on probability space $(\Omega,\mathcal{F},P)$. Prove that if $X_n\to X$ in probability then ${P_{{X_n}}} \Rightarrow {P_X}$ (weakly convergence, that is $\int {fd{P_{{X_n}}}} \to \int {fd{P_X}}$ for all $f\in C_b(\mathbb{R})$). Here ${P_X} = P \circ {X^{ - 1}}$, induced probability measure by $X$. Can someone help me?

• What is $C_b(\mathbb{R})$? – sinbadh Jan 15 '16 at 17:46
• $C_b(\mathbb{R})$ is the set of all bounded continuous functions on $\mathbb{R}$. – User3101 Jan 15 '16 at 23:02

For every $f\in C_b(\mathbb{R})$ one has that also $f(X_n)\to f(X)$ in probability. Since $f$ is bounded, the sequence $(f(X_n))_{n\geqslant 1}$ is uniformly integrable and hence $f(X_n)\to f(X)$ in $L^1$. This ensures that $\mathrm{E}[f(X_n)]\to\mathrm{E}[f(X)]$.