the Cameron-Martin theorem
/ CAM-er-on MAR-tin /
If you shift a Gaussian measure by a fixed vector, do you get an equivalent measure (one with a density relative to the original) or a totally different, mutually singular one? The Cameron-Martin theorem answers this completely: there is exactly one special subspace of admissible directions — the Cameron-Martin space H — along which shifting only re-weights the measure, and it even gives the explicit Radon-Nikodym density. Outside H the shifted measure is singular. This is the quasi-invariance of Gaussian measure, and it is the Gaussian replacement for the translation-invariance that Lebesgue measure has but no infinite-dimensional measure can.
Let mu be a centered Gaussian measure on a Banach space E, and let H be its Cameron-Martin space — the RKHS of the covariance, sitting densely but with mu(H) = 0. For h in H, let T_h(x) = x + h be the shift, and mu_h := (T_h)_# mu the shifted measure. The theorem says mu_h is equivalent to mu, with density d mu_h / d mu (x) = exp( < h, x >~ - (1/2) ||h||_H^2 ), where < h, . >~ is the first-chaos (Paley-Wiener) random variable paired with h under the RKHS isometry, and ||h||_H is the Cameron-Martin norm. Conversely, if h is NOT in H then mu_h and mu are mutually singular. So the admissible shifts are precisely the RKHS, and the density is a Gaussian exponential tilt by the paired chaos variable. For Brownian motion this is Girsanov's theorem in its purest, drift-only form: shifting paths by a finite-energy function h reweights Wiener measure by exp( integral h' dB - (1/2) integral h'^2 dt ).
Why it matters: Cameron-Martin underlies Girsanov and the whole change-of-measure technology for diffusions, the integration-by-parts formula of Malliavin calculus, large deviations (Schilder's theorem, where the rate function is (1/2)||h||_H^2), and the very notion of 'differentiating in the H directions.' The non-negotiable subtlety: admissibility is a knife-edge. Almost every direction is forbidden — H is a measure-zero subspace — so the same shift that is harmless in R^n is catastrophic (singular) in infinite dimensions unless it happens to have finite Cameron-Martin norm.
On Wiener space, shift Brownian motion by h with h(t) = integral_0^t g(s) ds, g in L^2 (so h is in the Cameron-Martin space). Cameron-Martin says the law of B + h is equivalent to that of B, with density exp( integral_0^1 g dB - (1/2) integral_0^1 g^2 dt ). Shifting by a non-finite-energy function (e.g. a constant nonzero shift at every time, h(t) = c) is forbidden: it yields a singular measure.
Shifting along a finite-energy (Cameron-Martin) path reweights Wiener measure by a Girsanov exponential; other shifts are singular.
Admissible shifts form a measure-zero subspace: almost all directions make the shifted Gaussian measure singular, so quasi-invariance holds ONLY along the Cameron-Martin space.