The space $L^2(\mathcal{F}_T, P)$ is used in my textbook, but does not seem to be defined anywhere that I can find (and is ommited in the symbols table at the end of the book). Here, $P$ is a probability measure, and $\{\mathcal{F}_t\}$ is a filtration.
As I understand it, a function $f \in L^2 (P)$ if $(\int f^2 dP)^{1/2} < \infty$
The notation $L^2(\mathcal{F}_T, P)$ is used in the context of Ito integrals from $S < t < T$ where $\mathcal{F}_t$ is the filtration generated by the Brownian motion process.
So what does $L^2(\mathcal{F}_T, P)$ mean?
I am guessing it means that $\int f^2 dP < \infty$ and $f(t,\omega)$ is $\mathcal{F}_t$-adapted. But it's never stated explicitly.