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226 XIII INTEGRATION
(13.21.9) A ^-measurable function h is v-integrable if and only if
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This follows from (13.21.8) and (13.9.13).
(13.21.10) Let A be a ^-measurable set in X x Y.
(i) The set M of all x e X such that the section A(x) is not \i-measurable is
^-negligible', the function jth-»/^*(A(.x)) is ^-measurable; and |
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*(A) = J V
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In particular, ifA(x) is ^-negligible except on a ^-negligible set of values ofx,
then A is ^-negligible.
(ii) If A is v-integrable, then the set of all x e X such that A(x) is not
\L-integrable is ^-negligible; the function x\-*/j,(A(x)) (which is defined almost
everywhere with respect to X) h X-integrable; and v(A) = \fi(A(x)) dk(x).
These assertions are particular cases of (13.21.6), (13.21.8) and (13.21.7).
(13.21.11). Letf(resp.g) be a mapping ofX (resp. Y) into [0, +00]. With
the convention of (1 3.11 ) for products, we have |
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(13.21.11.1) /(xfoOO d%x) dp(y) = f */(x)
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\J
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By virtue of (13.21.4), we have
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jj /W000 d%x) dn(y) ^ f * dl(x) Ff(x)g(y) dn(y).
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On the other hand, for each x e X we have
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(
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and
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x) J *f(x)g(y) dn(y) = f *( (
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g(y)
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