So I am reading an introductory script on stochastic analysis in Hilbert spaces and there is a step in the proof of "Gaussian measures have trace-class covariance" that I don't understand:
We are working with a Gaussian measure $\mu$ on a separable Hilbert space $H$, i.e., a probability measure $\mu$ on $(H,\mathcal{B}(H))$ such that for some mean $m \in H$ and covariance $Q \in \mathcal{L}(H)$ the pushforward measures satisfy $(\langle h,\cdot\rangle_H)_{\#\mu} \sim \mathcal{N}(\langle m,h\rangle_H, \langle Q h,h\rangle_H)$ for all $h \in H$. We now want to prove that $Q$ is trace class, i.e., we need to show for some complete orthonormal system $(e_i)_{i \in \mathbb{N}}$ in $H$ that $\sum_{i \in \mathbb{N}} \langle Q e_i,e_i\rangle_H<\infty$.
The proof idea in my script is to restrict ourselves to the case $m=0$ for simplicity and then use the fact that the neighborhoods $(U_{1/2}(e_i))_{i \in \mathbb{N}}$ are disjoint and contained in $U_2(0)$, which has finite measure under $\mu$ (since $\mu$ is a probability measure). Thus, we obtain by $\sigma$-additivity that $\sum_{i \in \mathbb{N}} \mu(U_{1/2}(e_i))< \infty$. According to the script, the statements now follows since $\langle Q e_i,e_i\rangle_H \sim \mu(U_{1/2}(e_i))$, but I can't seem to figure out why these measures and the variances are comparable. Can somebody give me a hint please?
