Assume that $P_n$ is a sequence of probability measures on $D[0,1]$ with the Skorohod topology. Assume that they converge weakly to a measure $P$. Let $C$ be the space of continuous functions on $[0,1]$, if $P_n(C)=1$, is it then the case that $P(C)=1$?
2026-02-23 09:05:17.1771837517
Will weak convergence of continuous functions give a continuous function?
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$P(C)\ge \limsup\limits_{n\to\infty} P_n(C)=1$, since $C$ is a closed set in $D$.