I'm working on Di Nunno, Øksendal, and Proske's book on Malliavin Calculus and stuck in the problem below. I wrote a possible solution here and discussed it afterward.
Problem. Let $u(t)$, $0 \le t \le T$, be a measurable stochastic process such that $$ \mathbb{E} \left[ \int_0^T u^2(t) ~\mathrm{d}t \right] < \infty $$
Show that there exists a sequence of deterministic measurable kernels $f_n(t_1, \ldots, t_n, t)$ on $[0,T]^{n+1}$ $(n \ge 0 )$, with $$ \int_{[0,T]^{n+1}} f_n^2 (t_1, \ldots, t_n, t) ~\mathrm{d}t_1 \cdots \mathrm{d}t_n \mathrm{d}t < \infty $$ such that all $f_n$ are symmetric with respect to the variables $t_1, \ldots, t_n$ and such that $$ u(t) = u(\omega, t) = \sum_{n=0}^\infty I_n(f_n(\cdot, t))(\omega), \quad \omega \in \Omega, ~t \in [0,T] $$ with convergence in $L^2(\mathbb{P} \times \lambda)$.
Hint: Consider approximations of $u(t)$, $t \in [0,T]$, in $L^2(\mathbb{P} \times \lambda)$ of the form $\sum_{i=1}^m a_i(\omega) b_i(t)$, $m = 1, 2, \ldots$, where $a_i \in L^2(\mathbb{P})$ and $b_i \in L^2([0,T])$.
Using the hint, consider the approximations of $u(t)$ $$ \varphi_m(t) = \sum_{i=1}^m a_i(\omega) b_i(t) $$ with $a_i \in L^2(\mathbb{P})$ and $b_i \in L^2([0,T])$.
By the Wiener-Itô Chaos Expansion, for each $a_i$ there exists a unique sequence of functions $g_n^{(i)} \in \tilde{L}^2 ([0,T]^n)$ such that $$ a_i = \sum_{n=0}^\infty I_n(g_n^{(i)}) $$
Then, $$ \varphi_m(t) = \sum_{i=1}^m \sum_{n=0}^\infty b_i(t) I_n(g_n^{(i)}) $$
Just opening the iterated integral gives $$ \varphi_m(t) = \sum_{i=1}^m \sum_{n=0}^\infty b_i(t) \int_{[0,T]^n} g_n^{(i)} (t_1, \ldots, t_n) ~\mathrm{d}W(t_1) \cdots \mathrm{d}W(t_n) $$
Interchanging the series and the integral, $$ \varphi_m(t) = \sum_{n=0}^\infty \int_{[0,T]^n} \sum_{i=1}^m b_i(t) g_n^{(i)} (t_1, \ldots, t_n) ~\mathrm{d}W(t_1) \cdots \mathrm{d}W(t_n) $$
Taking $m \to \infty$, $\varphi_m \to u$. We define $$ f_n(t_1, \ldots, t_n, t) = \sum_{n=0}^\infty b_i(t) g_n^{(i)} (t_1, \ldots, t_n) $$ and obtain $$ u(t) = \sum_{n=0}^\infty b_i(t) \int_{[0,T]^n} f_n (t_1, \ldots, t_n, t) ~\mathrm{d}W(t_1) \cdots \mathrm{d}W(t_n) = \sum_{n=0}^\infty I_n(f_n(\cdot, t)) $$
There are two immediate problems here:
Can I exchange the series and the integral?
What guarantees the existence of $f_n$, i.e. the existence of that limit?
Are there any other problems? Is it possible to fix them?
I also tried to emulate the proof for the Wiener-Itô Chaos Expansion. Should I do it on the product $a_i b_i$? Is this on $L^2(\mathbb{P})$ to apply Itô Representation Theorem?