Abstract
A submetric space is a topological space with continuous metrics, generating a metric topology weaker than the original one (e.g. a separable Hilbert space with the weak topology). We demonstrate that on submetric spaces there exists a theory of convergence in probability, in law etc. equally effective as the Probability Theory on metric spaces. In the theory on submetric spaces the central role is played by a version of the Skorokhod almost sure representation, proved by the author some 25 years ago and in 2010 rediscovered by specialists in stochastic partial differential equations in the form of “stochastic compactness method”.
Summary
This paper introduces the concept of submetric spaces — topological spaces (X, tau) that contain a metric topology tau_m weaker than tau — and shows that probability theory on such spaces is as effective as on metric spaces. The key examples are separable Banach spaces with the weak topology, and the Skorokhod space D with Jakubowski’s S-topology. Both are non-metrizable but submetric.
The central tool is the Skorokhod almost sure representation for subsequences (Theorem 3.1): on a space with property CCSP (a countable family of continuous functions separating points), every uniformly tau-tight sequence of probability measures contains a subsequence admitting a Skorokhod representation. This is a sequential version of Prokhorov’s theorem that does NOT require metrizability.
The crucial distinction is between tau-tightness (mass on tau-compact sets) and d-tightness (mass on d-bounded sets). Fernique’s example (p. 142) shows these are genuinely different: on a separable Hilbert space H with weak topology tau_w, there exist sequences X_n converging weakly in distribution to 0 but with P(||X_n|| > R) → 1, so no subsequence is uniformly tau_w-tight. The escape of mass to infinity in the norm sense is invisible to the weak topology.
Key Contributions
- Definition of submetric spaces (Definition 4.1) and proof that the sequential Prokhorov theorem extends to them (Theorem 4.2)
- Identification of property CCSP as the key requirement for sequential compactness arguments in non-metrizable spaces
- Fernique’s example showing tau-tightness and d-tightness are genuinely different in infinite dimensions
- Framework for convergence in probability and in law on submetric spaces (Section 5), including definition of L_0(Omega : (X, tau)) as the space of random elements
Methodology
The approach builds on a single key observation: on tau-compact sets K, the original topology tau and any compatible metric topology tau_d coincide (since K is compact and tau_d is Hausdorff, the identity map (K, tau) → (K, tau_d) is a homeomorphism). Therefore tau-compact sets are metrizable, even if X is not. This means sequences in tau-compact sets behave like sequences in metric spaces.
Key Findings
- CCSP ⇒ sequential Prokhorov (Theorem 3.1): uniform tau-tightness gives subsequences with Skorokhod representations, without metrizability
- Submetric ⇒ CCSP (immediate from Definition 4.1): the countable separating family {f_i} defines the continuous metric d(x,y) = sum 2^{-i} |f_i(x) - f_i(y)| / (1 + |f_i(x) - f_i(y)|)
- tau-compact = tau_d-compact on submetric spaces: compactness is a topological invariant independent of which compatible metric is used
- Convergence in distribution vs in probability: both can be defined on submetric spaces via the CCSP metric, and the theory parallels the metric space theory
- Tightness is essential: Remark 3.3 notes that even spaces with CCSP need not be Prohorov (uniform tightness cannot be dropped)
Important References
- A Non-Skorohod Topology on the Skorohod Space — Jakubowski (1997), the original S-topology paper where Theorem 3.1 was first proved
- The almost sure Skorokhod representation for subsequences in nonmetric spaces — Jakubowski (1998), the Theory of Probability & Applications version
- Convergence of Probability Measures — Billingsley, the classical reference for Skorokhod representation on metric spaces