Let $\mathbf{D} := D([0,1]; \mathbb{R}^d)$ be the Skorokhod space (equipped with the Skorokhod metric) of càdlàg functions, and let $X = (X_t)_{t \geq 0}$ be its canonical process. The space of probability measures on $\mathbf{D}$ is denoted $\mathcal{P}(\mathbf{D})$. It is well known that weak convergence/topology on $\mathcal{P}(\mathbf{D})$ is metrizable. I'm now wondering the following:

If we equip $\mathcal{P}(D)$ with the topology $\mathcal{T}_w$ of convergence of finite-dimensional distributions, that is, the coarsest topology making the maps of the form

$$ \mathcal{P}(\mathbf{D}) \ni \mathbb{P} \mapsto E_{\mathbb{P}}[f(X_{t_1},\ldots,X_{t_n})] \in \mathbb{R}$$

continuous for all $n \in \mathbb{N}$, all $0 \leq t_1 < \ldots < t_n \leq 1$ and each bounded, continuous $f : \mathbb{R}^n \rightarrow \mathbb{R}$, is then $\mathcal{T}_w$ metrizable?

  • $\begingroup$ Is there a countable basis of neighborhoods at each point? $\endgroup$ – Anthony Quas Mar 25 at 16:21
  • $\begingroup$ Do you mean for $\mathbf{D}$? If so, then yes as $\mathbf{D}$ is polish. $\endgroup$ – vaoy Mar 25 at 16:46
  • $\begingroup$ I mean in the topology $\mathcal T_w$, does each measure have a countable basis of neighborhoods containing it? $\endgroup$ – Anthony Quas Mar 25 at 17:21
  • $\begingroup$ It seems like the question boils down to being able to define $\mathcal{T}_w$ with only say rational or dyadic values for the $t_i$'s. $\endgroup$ – Abdelmalek Abdesselam Mar 25 at 21:38

Convergence of FDDs implies convergence in the "convergence in measure" sense, that is in the space L^0 as defined in e.g. https://en.wikipedia.org/wiki/Lp_space#L0,_the_space_of_measurable_functions . That is an often-rediscovered theorem. Of course L^0 is metrizable. Informally "the two topologies are almost the same, on D", so perhaps whatever you want to do could be done within L^0. See e.g. my old paper "Stopping times and tightness. II. Ann. Probab. 17 (1989), 586–595" for one use of this device.


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