# Stochastic Fubini (Da Prato & Zabcyzk)

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 $$(x,t,\omega)$$?