I want to expand on the question already posed here: Does convergence in probability w.r.t. a topology make sense? . Suppose we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a topological space $(E,\tau)$, which is a measurable space once endowed with the Borel $\sigma$-algebra $\mathcal{B}(\tau)$. Given a sequence $\{X_n\}$ of $E$-valued random variables, we can then try to define convergence in probability for such a sequence. In the aforementioned question, it is proposed to use the characterization "$X_n\to X$ in probability iff every subsequence contains a subsequence converging $\mathbb{P}$-a.s.", since the notion of convergence only relies on the topology (i.e. no metric structure is required). However, if the space $E$ is a bit more structured, like being a topological vector space, then one could say that "$X_n\to X$ in probability iff $X_n-X\to 0$ in probability", where by the latter we mean that \begin{equation} \mathbb{P}(X_n-X\in U)\to 1\ \text{ as }\ n\to\infty\ \text{ for every } U \text{ neighbourhood of } 0 \end{equation} So a first possible question is: in this case, under which assumptions on $\tau$ the two notions are equivalent? I know that under some assumptions the topological vector space becomes metrizable and so the answer in that case becomes trivial, so I'm referring to "the other cases".

Take for instance the following case: $E$ is an infinite dimensional, countable Hilbert space. On $E$ we have the natural topology induced by the inner product, but we can take instead the weak topology $\sigma(E,E^\ast)$ induced by such inner product. Then the weak topology is not metrizable, but it is locally metrizable (i.e. when restricted to closed balls $B(0,R)$) and the Borel $\sigma$ algebra generated is the same generated by the strong topology. So in this case, both of the above definitions are meaningful; do they coincide? Are there any other references in the literature to this, or similar definition, than those already given in the aforementioned question?


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