If D is any scalar-valued measurable process, we have
D E L (X) if and only if E(f R IdiX! ) < -. In this case,
E(f 2 dX ) is defined,
(4.2.1) uX(f) = E(f 2zdXz), and
('4.2.2) | |j( ) E(fR 2 dzd jX )
i.e., |uX| X 1 |.
Proof. For M E M, the integral E(f 21MdXz) is defined; to see
this, we use a monotone classes argument. More precisely, let
H = {M E M: E(fR lMdXz) Is defined}. We will show that H is an
algebra, is closed under monotone convergence, and contains a semiring
generating M; we then conclude from the monotone class theorem that
H : M, hence H = M.
H is an algebra. Let A,B E H; show A kB, An IB, AC E H. First of
all, f 21A(w)dX(w) exists P-a.s. as well as f 21B(w)dX (w).
Then 1 A(w) 1 (w) is dXz(w)-integrable almost surely, i.e.,
f 21A(w) A 1B(w)dX(w) = f 21AB(w)dXz(w) exists P-a.s. Moreover,
+ -+
R2nAB(w)dXz () f21AnB(w)dlX z(w) fR21A(w)d X z(w); hence
w R21AnB(w)dXz(w) is P-integrable, i.e., E(f 1 BdX ) exists,