Two Gaussian measures are either mutually singular or equivalent. This dichotomy was first discovered by Feldman and Hajek (independently). We give a simple, almost formal, proof of this result, based on the study of a certain pair of functionals of the two measures. In addition we show that two Gaussian measures with zero means and smooth Polya-type covariances (on an interval) are equivalent if and only if the right-hand slopes of the covariances at zero are equal. The H and J functionals*

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... probability measures μ 0 and μ x on a space (Ω, &) are called mutually singular (μ 0 _L μd if there is a set ΰe & for which μ o (B) -0 and μ λ {Ω -B) = 0. The measures are called mutually equivalent (μ 0 ~ μ ± ) if they have the same zero sets, i.e., μ o (B) = 0 if and only if μ^B) = 0. Setting μ = μ 0 + μ x we may define the Radon-Nikodym derivatives, (1.1) X Q = dμ o /dμ , X, = dμjdμ .