An application of Lebesgue's dominated convergence theorem

Question

Let

• $U,H$ be separable $\mathbb R$-Hilbert spaces
• $(e^n)_{n\in\mathbb N}$ be an orthonormal basis of $U$
• $(\Omega,\mathcal A,\operatorname P)$ be a probability space
• $\mathcal F\subseteq\mathcal A$ be a $\sigma$-algebra on $\Omega$
• $\mathfrak L(U,H)$ denote the space of bounded linear operators from $U$ to $H$
• $X:\Omega\to\mathfrak L(U,H)$ be strongly $\operatorname P$-Bochner integrable and $Y\in\mathcal L^1(\operatorname P,U)$ with $$\left\|X\right\|_{\mathfrak L(U,\:H)}\left\|Y\right\|_U\in\mathcal L^1(\operatorname P)\tag1$$

Since $(e^n)_{n\in\mathbb N}$ is an orthonormal basis of $U$ and $X(\omega)$ is continuous, we obtain $$X(\omega)Y(\omega)=\sum_{n\in\mathbb N}\langle Y(\omega),e^n\rangle_UX(\omega)e^n\tag2$$ in $H$ for all $\omega\in\Omega$. Assuming $X$ is independent of $\mathcal F$ and $Y$ is $\mathcal F$-measurable, I would like to conclude that $$\sum_{n=1}^N\operatorname E\left[\langle Y,e^n\rangle_UXe^n\mid\mathcal F\right]\xrightarrow{N\to\infty}\operatorname E\left[XY\mid\mathcal F\right]\;\;\;\operatorname P\text{-almost surely}\;.\tag3$$ This would hold by Lebesgue's dominated convergence theorem, if we could show that $$\left\|\sum_{n=1}^N\langle Y,e^n\rangle_UXe^n\right\|_H$$ can be bounded by a $\mathcal L^1(\operatorname P)$-element uniformly with respect to $N\in\mathbb N$. Can we do that? And if not, which additional assumption do we need?

Answer 1

Since $X(\omega)\in\mathfrak L(U,H)$ for all $\omega\in\Omega$, we obtain $$\left\|\sum_{n=1}^N\langle Y,e^n\rangle_UXe^n\right\|_H^2\le\left\|X\right\|_{\mathfrak L(U,\:H)}^2\left\|\sum_{n=1}^N\langle Y,e^n\rangle_Ue^n\right\|_H^2\xrightarrow{N\to\infty}\left\|X\right\|_{\mathfrak L(U,\:H)}^2\left\|Y\right\|_H^2\tag4$$ by Parseval's identity. The right-hand side of $(4)$ is in $\mathcal L^1(\operatorname P)$ by assumption.