I've been studying Da Prato & Zabcyzk's text on SPDE, and there's a part of their proof of the stochastic Fubini theorem that I don't quite get. The setting is that $\Phi: (\Omega_T\times E, \mathcal{P}_T \times \mathcal{B}(E))\to (L_2^0, \mathcal{B}(L_2^0))$, such that $$ \int_E \|\Phi(\cdot, \cdot,x)\|_T\mu(dx)< \infty $$ where $\mu$ is a finite positive measure and the $T$ norm corresponds to $$ \|\Phi\|_T^2 = \mathbb{E}\int_0^T\|\Phi(s)\|_{L_2^0}^2 ds $$

The authors have an intermediate result, Prop 4.34 in the 2014 edition, that $\Phi$ can be approximated by a sequence $\Phi_n$ of analogously measurable processes that can be written as simple processes and converge in the sense of $$ \int_E \|\Phi(\cdot, \cdot,x)-\Phi_n(\cdot, \cdot,x)\|_T\mu(dx) \to 0. $$

I find myself unsure of the proof of this intermediate result in that the authors first assert the existence of a sequence $\Psi_n$ (not neccessarily simple) which are bounded and converge in the above sense. It is clear from the above assumption that $\mu$-a.e., there are processes $\Psi_n(\cdot,\cdot,x)$ which converge in the $T$ norm. This follows from a Prop 4.22 of the text, and it is clear that they can be taken such that $$ \|\Psi_n(\cdot,\cdot,x)\|_T \leq 2\|\Phi(\cdot,\cdot,x)\|_T. $$ How can I infer that this sequence $\{\Psi_n\}$ is then bounded as a mapping over all three arguments $(x,t,\omega)$?


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