I am reading the book "An invitation to the Algebra of Canonical Commutation Relations" by Denes Petz. It is freely available for download here. In Chapter 9, he defines the Lebesgue measure on a finite dimensional symplectic space. To set up the notations, let $(H,\sigma)$ be $2n$-dimensional symplectic space and $\,dg$ stand for the Lebesgue measure induced on $H$ from the standard symplectic space $\mathbb{R}^{2n}$. Let $CCR(H,\sigma)$ denote the CCR Algebra over $(H,\sigma)$ generated by the elements $\{W(g):g\in H\}$ satisfying the Weyl commutation relations. I have a problem in understanding the following. He writes,
"Let $\phi$ be a Fock state on $CCR(H, \sigma)$. The integral
\begin{equation} \pi^{-n}\int_H \phi(W(g))W(g) \,dg = P \end{equation} is norm convergent and defines an element $P$ of $CCR(H,\sigma)$."
My view: I understand that the integral under consideration is the Bochner integral and all I need is to find a sequence of simple functions converging in norm to $\phi(W(g))W(g)$. But I am unable to produce such a sequence.
