Let H be a separable Hilbert space. Let L be a Hilbert-Schmidt operator from H to H, such that its image is dense in H. This allow one to take the basis of eigenfunctions of $L$: $\{\phi_i\}_{i=1}^\infty$ (such that $L\phi_i=\lambda_i\phi_i$, $\lambda_i$ being the eigenvalues of $L$), and define a projector sequence $\{P_n\}_{n=1}^\infty$: $P_nh=\sum_{i=1}^n[h,\phi_i]\phi_i$. Moreover, let $\mu_G$ be the Gaussian measure on H (which is finite additive).
The sequence of measure defined as $\mu_n=\mu_G\circ(LP_k)^{-1}$ converges weakly to a countably additive tight measure $\nu_L$, which we call the measure induced by $L$.
If K is a compact set such that $\nu_L(K)>1-\epsilon$, what can I say about $\mu_n(K)$? Is there some condition such that also $\mu_n(K)>1-\epsilon$ too?
