we have f is d(gx)-integrable on I and fIfnd(gx) -
J fd(gx). Now, (4.3.1) holds for each n, so passing to
limits as before, (4.3.1) holds for f as well.
Now, let S be the family of sets of the form R = ((s,t),(s',t')]
( + R2 Since the rectangles ((s,t),(s',t')] form a semiring
2 a
generating the Borel o-field on R2, S is a semiring generating B(R ).
Let, then, C be the family of indicators of sets of S. To complete
the monotone class argument, we must show that CCH and that C is
closed under multiplication, as H then contains all bounded functions
measurable with respect to o(C) = B(R).
C is closed under multiplication, as 1 1 = 1 and S is
1 2 1 2
a semiring, so R R2 E S => 1R R2 E C. Now we show that (4.3.1)
1 2
holds for f = 1R, R E S. We have
xf1i dg = xf1Rildg = x(g(Rn I))
(Again g(-) refers to the measure dg.) Since Ix(g(R nI))j
|x.* gl(R fI) S |xI-jg|(I) < -, 1R is dg-integrable and d(gx)-
integrable. Also, fIX1Rdg = fx1 n dg = x(g(R nI)), and
JI1Rd(gx) = J1ROid(gx) = (gx)(R I) = x(g(Rn I)), hence
xfI1Rdg = fix1Rdg = IJ1Rd(gx), which is (4.3.1), so C H. This
completes the proof for f bounded, Ig (I) < -.
Assume now, Igl(I) < -, f dg-lntegrable on I (not necessarily
bounded). There exists a sequence (f ) of bounded functions
n