Let $(\Omega, \mathcal F\_1, {\mathbb P}, \mathbb F = \{\mathcal F\_t\}\_{0\le t \le 1})$ is a
filtered probability space. Let $L^2\_{\mathbb F}$ be a collection of all $\mathbb F$ progressive measurable 
1-d processes $x$, such that
$$\mathbb E [\int\_0^1 x^2(s) ds ] <\infty.$$
Define
$$\langle x, y \rangle = \mathbb E \int\_0^1 x(s) y(s) ds.$$
My question is


[Q.] Is $L^2\_{\mathbb F}$ a Hilbert space with operator above defined inner product?


The answer seems to be positive. The only problematic proof is about the completeness. That is,
if $x\_n$ converges to $x$ in the sense of
$\mathbb E [\int\_0^1 |x\_n-x|^2(s) ds ] \to 0,$
is $x$ also progressively measurable?



Does $L^2$ progressive measurable processes form a Hilbert space?
The answer is indeed yes, as every adapted measurable real-valued process admits a progressively measurable modification. This is classical Theorem due to Paul-André Meyer (cf. Paul A. Meyer. Probabilities and Potential, Blaisdell 1966; Theorem T46, p. 68)


