we have f 2A (w)dX(w) A (w)dX (w) P-a.s. Then, the maps
R R
latter is integrable as we saw above, so by monotone convergence we
have that E(f 2A"dX) E(f A dX ). Since t(An) > y(A) and (4.4.1)
R z z R z z
holds for each n, (4.4.1) holds in the limit as well. Also, H is
closed under uniform convergence: let A be bounded, measurable, An a
sequence from H with An A uniformly. By Lebesgue p(A) (A).
Also, for almost all w, X(s) is a bounded positive measure, so
f 2An(w)dX) 2 (w)dX (w) P-a.s. (by Lebesgue again) so by
R z z R2 z z
Lebesgue the map w f 2A (w)dX (w) Is P-integrable (since each
R z
An E H) and E( dX) E( AzdX ). Since (4.4.1) holds for each
n, it holds in the limit as well. Finally, let An be uniformly
bounded, A tA, A c H for all n. As above, using a double application
of the monotone convergence theorem this time, we have E(f 2AdX )
SE(f 2AzdXz), and p(A ) p(A), and we conclude as above.
To complete the monotone class argument, let C be the class of
processes of the form Y(w) I[0,](z). We already know C H; it is
easy to see (taking Y indicators of sets of F) that o(C) M.
Finally, C is closed under multiplication; in fact,
(Y1() I ,u](z))(Y2 I ,v]( )) = ( )y2(w)) I[ uv (z)
c C, and the monotone class argument is done. This completes the