I got stuck with the following problem while reading Section 10.18 of 'Lectures on von Neumann algebras' by Strătilă and Zsidó.
Problem: Let $\varphi$ be a normal weight on a von Neumann algebra $\mathscr{M}$. Let $\{\sigma_t\}_{t\in\mathbb{R}}$ be a group of automorphisms of $\mathscr{M}$. For $a\in\mathscr{M}^+$, we define $a_n=\sqrt{n/\pi}\int_{-\infty}^{+\infty}e^{-nt^2}\sigma_t(a)\,dt,\;n\in\mathbb{N}$. Then show that $\varphi (a_n)=\sqrt{n/\pi}\int_{-\infty}^{+\infty}e^{-nt^2}\varphi (\sigma_t(a))\,dt,\;n\in\mathbb{N}$.
Definition: A weight $\varphi$ on a von Neumann algebra $\mathscr{M}$ is normal if there exists a family $\{\varphi_i\}$ of $w$-continuous (i.e. continuous with respect to the ultraweak topology on $\mathscr{M}$) positive forms on $\mathscr{M}$ such that $\varphi (a)=\sum_i\varphi_i (a),\;a\in\mathscr{M}^+$.
To solve the above problem, it is enough to show that $\sum_i\sqrt{n/\pi}\int_{-\infty}^{\infty}e^{-nt^2}\varphi_i(\sigma_t(a))\,dt=\sqrt{n/\pi}\int_{-\infty}^{+\infty}e^{-nt^2}\sum_i\varphi_i(\sigma_t(a))\,dt$, and it almost seems like an application of Tonelli's theorem, but here the summation $\sum_i$ can be an uncountable sum, which being not $\sigma$-finite, we can not apply Tonelli's theorem.
Note that I have changed the actual statement given in the book to reduce some technicality. To see the actual statement I got stuck in, see the highlighted part of this attachment. Please let me know if you need more details. Thanks in advance for any help or suggestion.
It's easier to use directly that $\varphi$ is normal. The sequence $$ b_k=\sqrt{\frac n\pi}\int_{-k}^ke^{-nt^2}\,\sigma_t(a)\,dt $$ is increasing on $k$, and $\lim_kb_k=a_n$. Then the normality of $\varphi$ gives you $$ \varphi(a_n)=\lim_k\varphi(b_k)=\sqrt{\frac n\pi}\int_{-\infty}^\infty e^{-nt^2}\,\varphi(\sigma_t(a))\,dt. $$