Let $f:X\to Y$ be a Hilbert space valued function, with $Y$ a separable Hilbert space and $X$ a measurable space. Furthermore, assume that $$ \int \|f(x)\|_Y \mu(dx)<\infty, $$ for $\mu$ is a positive, finite, measure on $X$ and an associated $\sigma$-algebra.
At each value of $x$, $f(x)$ is to be approximated by simple functions of the form $$ f_n(x) = \sum_{j=1}^{J_n} c_j^n e_j 1_{A^n_j}(x) $$ where $e_j$ are an orthonormal basis of $Y$, $A^n_j$ are measurable subsets of $X$ and $c_j^n$ are scalar coefficients. The goal is for these to satisfy $$ \lim_{n\to \infty} \int\|f(x) - f_n(x)\|_Y\mu(dx)\to 0 $$
Two questions:
- Can we say anything about the measurability, in the $x$ argument, of the $\{f_n(x)\}$?
- Can we say anything about the boundedness (in the $Y$ norm) of $\{f_n(\cdot)\}$ over both $x$ and $n$?
I am attempting to reformulate a more complicated problem (originally asked here into what I believe are its essential elements.