the abstract Wiener space
/ VEE-ner /
The abstract Wiener space is Leonard Gross's distillation of 'what a Gaussian measure on an infinite-dimensional space really is.' It packages the three players — a Hilbert space of admissible directions, a bigger Banach space the measure actually lives on, and the Gaussian measure itself — into one canonical triple, and explains the strange fact that the Gaussian measure sits on a space strictly LARGER than the Hilbert space you start from.
An abstract Wiener space is a triple (H, B, mu) where H is a separable Hilbert space (the Cameron-Martin space), B is a separable Banach space, the inclusion H -> B is continuous with dense image, and mu is the Gaussian measure on B whose covariance is induced by the inner product of H. Gross's theorem is the construction: the standard Gaussian 'measure' on H itself (the one with covariance equal to the identity, i.e. independent N(0,1) in every coordinate of an orthonormal basis) does NOT exist as a countably-additive measure on H when dim H = infinity — the total mass would have to spread over infinitely many independent unit-variance directions and escapes to infinity. But it DOES exist if you weaken the norm: choose a measurable norm on H (in Gross's sense, weaker than the Hilbert norm), complete H in that norm to get B, and the cylinder-set Gaussian extends to a genuine Borel Gaussian measure mu on B. Then mu(H) = 0: the measure lives on B but not on the Hilbert space that generated it.
Why it matters: this is the rigorous home for Wiener measure (H = Cameron-Martin space of finite-energy paths, B = C[0,1], mu = Wiener measure), for white noise, and for the whole machinery of Malliavin calculus, Cameron-Martin shifts, and infinite-dimensional analysis (the Ornstein-Uhlenbeck operator and log-Sobolev inequality are stated on (H, B, mu)). The key honest point it makes precise: the gap between H and B is unavoidable in infinite dimensions — you cannot put a non-degenerate Gaussian on a Hilbert space with covariance the identity, so the Cameron-Martin space is always a strict, measure-zero subspace of where the randomness actually lives.
Classical Wiener space is the canonical example: H = { finite-energy paths h, h(0) = 0, integral h'^2 < infinity } with the energy inner product, B = C[0, 1], and mu = Wiener measure (law of Brownian motion). Brownian paths live in C[0, 1] (continuous) but never in H (they are not differentiable), so mu(H) = 0 — the picture (H, B, mu) made concrete.
Classical Wiener space: finite-energy paths H sit densely but with measure zero inside C[0,1], where Wiener measure actually lives.
No non-degenerate Gaussian measure with identity covariance exists on an infinite-dimensional Hilbert space; you must pass to a larger Banach space B, and then mu(H) = 0.