Let $\mu$ be a centered Gaussian measure on a separable infinite-dimensional Hilbert space $H$. For every $\delta>0$ does there exist a convex and compact set $C_{\delta}\subseteq H$ such that: $$ \lim\limits_{\delta\downarrow 0}\,\mu(C_{\delta}) =1 $$ and there are $N(\delta)$-dimensional linear subspaces $X_{\delta}\subset X$ such that $$ C_{\delta}\subseteq X_{\delta}? $$
In other words, can I approximate Gaussian measures on Hilbert spaces by some "finite-dimensional" analogue?
This is not true in general. E.g., suppose that $H=\ell^2$ and $\mu$ is the distribution of the random vector $$G:=\sum_{k=1}^\infty \frac{Z_k}k\,e_k,$$ where the $Z_k$'s are independent standard normal random variables and the $e_k$'s form the standard basis of $\ell^2$.
Take any finite-dimensional vector subspace $Y$ of $H=\ell^2$. Then $Y$ is contained in some vector subspace $X$ of $H=\ell^2$ of codimension $1$, so that $$X=\{x\in H\colon a\cdot x=0\}$$ for some nonzero $a\in H$, where $\cdot$ is the inner product in $H=\ell^2$. Then $$a\cdot G=\sum_{k=1}^\infty \frac{Z_k}k\,a_k\sim N(0,s^2),$$ where $s^2:=\sum_{k=1}^\infty \frac{a_k^2}{k^2}>0$, so that $$\mu(Y)\le\mu(X)=P(G\in X)=P(a\cdot G=0)=0.$$ So, any finite-dimensional vector subspace $Y$ of $H$ has $\mu$-measure $0$.
