= Cameron-Martin space of a Gaussian measure
{c}
{title2=$E_\mu=\operatorname{Ran}\Sigma^{1/2}$}
For a centered <Gaussian measure> on a real <Hilbert space> with injective <covariance operator of a Gaussian measure> $\Sigma$, its Cameron-Martin space is $E_\mu=\operatorname{Ran}\Sigma^{1/2}$, with <inner product> $\langle h,g\rangle_{E_\mu}=\langle\Sigma^{-1/2}h,\Sigma^{-1/2}g\rangle$. In an <orthonormal basis> satisfying $\Sigma e_j=\lambda_je_j$, it consists of exactly the vectors $h$ for which $\sum_jh_j^2/\lambda_j<\infty$. It is complete in this stronger norm even if its range is not closed in the ambient norm. Its importance is that it describes precisely the translations that preserve the null sets of the <Gaussian measure>, as in the <Cameron-Martin theorem for a Gaussian measure>. For generalized <Gaussian white noise> over $L^2$, the corresponding Cameron-Martin space is <L2 space> itself.
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