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The definitions are from David Pollard, Convergence of Stochastic Processes, IV.1.1 p.65

Let $(\Omega,\mathcal{A},P)$ be any probability space. Let $(S,\mathcal{S})$ be a metric space with any $\sigma$-field $\mathcal{S}$.

Definition 1 We call $X:\Omega\to (S,\mathcal{S})$ random element in $(S,\mathcal{S})$ if $X^{-1}(\mathcal{S})\subset\mathcal{A}$

Definition 2 Let $X,X_1,X_2,\dots\colon\Omega\to (S,\mathcal{S})$ be random elements in $(S,\mathcal{S})$. We say that $X_n$ converges $(\mathcal{S})$-weakly to $X$ if $Pf(X_n)\to Pf(X)$ for every bounded, continuous and $(\mathcal{S},\mathcal{B}(\mathbb{R}))$-measurable $f\colon S\to \mathbb{R}$. (Hence for every $f$ for which the expression makes sense.)

Note that we can still call random elements $(\mathcal{S})$-weakly convergent, even if they are measurable with respect to an even larger $\sigma$-field than $\mathcal{S}$. We then restrict the class of possible "test"-functions $f$, which makes weak convergence "easier" to check. Hence the following question:

Question Let $\mathcal{B}$ be the Borel-field in $S$ and let $\mathcal{O}$ be the field generated by the open balls in $S$. Let $X_1,X_2,\dots$ be random elements in $(S,\mathcal{B})$ (in particular they are random elements in $(S,\mathcal{O})$ since $\mathcal{O}\subset\mathcal{B}$) that converge $(\mathcal{O})$-weakly to a random element $X$ in $(S,\mathcal{B})$. Does then $X_n$ converge $(\mathcal{B})$ weakly too? That is to say does $Pf(X_1)\to P(f(X))$ not only for all bounded, continuous and $(\mathcal{O},\mathcal{B}(\mathbb{R}))$-measurable functions $f:S\to\mathbb{R}$, but for every bounded and continuous functions $f$? (A such function is automatically $((\mathcal{O},\mathcal{B}(\mathbb{R}))$-measurable.)

Comment The question is, if the weaker kind of weak convergence does still make sense for random elements for that the normal (borel-field) notion of weak convergence could be defined

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  • $\begingroup$ and sorry for not showing effort of solving. i'm fully in thesis pressure and stumbled upon this question. i made some minor attempts, but they didn't work and weren't worth posting here $\endgroup$ – Bananach Apr 15 '13 at 14:06

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