Let $g:M'\to\mathbb R$ be bounded and continuous. The composition $h=g\circ f$ is bounded and measurable, and every discontinuity point of $h$ is a discontinuity point of $f$. Thus
$$
\mathbb P(X\in D_h)\leq\mathbb P(X\in D_f)=0.
$$
The <Portmanteau theorem> includes the null-discontinuity criterion: if $X_n\xrightarrow dX$ and a bounded measurable function is continuous at $X$ almost surely, then its expectations converge. Hence
$$
\mathbb E[g(f(X_n))]\longrightarrow\mathbb E[g(f(X))].
$$
Since this holds for every bounded continuous $g$, it proves the <continuous mapping theorem> conclusion $f(X_n)\xrightarrow d f(X)$.

Solution

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

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The Portmanteau theorem gives equivalent formulations of weak convergence, including inequalities on open and closed sets and convergence of expectations for every bounded measurable function whose discontinuity set has zero limiting probability.


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