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200 XIII INTEGRATION

The mapping /i-» j/dA so defined is clearly a C-linear form on the

complex vector space JS?£(X, |A|) which extends the given C-linear form A
on jf _C(X).

(13.16.3) J/"A /5 dwy complex measure on X, w toe A = /z • |A|, w/zere /z fr a
locally \X\-integrable function such that \h(x)\ = 1 almost everywhere with
respect to |A|.

Let A_l9 A₂ denote the real measures ^?A, ./A, respectively. Then [AJ <^ |A|
and |A₂| ^ m (13.3.7), so that A_t and A₂ are measures with base |A|, and
therefore the same is true of A. If A = h • |A|, it follows that |A| = \h\ - |A|
(13.13.5), which implies that \h(x)\ = 1 almost everywhere with respect
to |A| (13.15.3).

It follows now from the definition (13.16.2) and from (13.14.3) that, for
any A-integrable function/, we have

(13.16.4) f/dA= \fhdW
and hence, by virtue of (13.16.3) and (13.10.3),

(13.16.5) /dA

£ (
J

\f\d\X\.

A function is said to be ^-negligible (resp. ^-measurable) if it is |A|-negligible
(resp. | A| -measurable). Likewise for sets.

If A and /z are two complex measures and if /is both A-integrable and
/x-integrable, then / is (A + /i)-integrable, by virtue of (13.16.1) and the
inequality |A + iA S |A| + |/z| (13J.8).

If A is a complex measure, a function g is said to be locally X-integrable
if it is locally |A|-integrable. In that case, for every function /e 3ff_c(X), the

function #/is A-integrable, and as in (1 3.1 3) we see that/ h-» | gfdl is a meas-

ure, denoted by g - A. If A = h - |A| (13.16.3), we have jfy dX =jfgh d|A|, or

equivalently g - A = (gh) • |A|, so that the study of measures of the form g - A
is immediately reduced to the case where A is a positive measure. In particular,
it follows from (13.13.4) that

(13.16.6) l<rA| = |<7|-|A|.



200 XIII INTEGRATION The mapping /i-» j/dA so defined is clearly a C-linear form on the complex vector space JS?£(X, \|A\|) which extends the given C-linear form A on jf _C(X). (13.16.3) J/"A /5 dwy complex measure on X, w toe A = /z • \|A\|, w/zere /z fr a locally \X\-integrable function such that \h(x)\ = 1 almost everywhere with respect to \|A\|. Let A_l9 A₂ denote the real measures ^?A, ./A, respectively. Then [AJ <^ \|A\| and \|A₂\| ^ m (13.3.7), so that A_t and A₂ are measures with base \|A\|, and therefore the same is true of A. If A = h • \|A\|, it follows that \|A\| = \h\ - \|A\| (13.13.5), which implies that \h(x)\ = 1 almost everywhere with respect to \|A\| (13.15.3). It follows now from the definition (13.16.2) and from (13.14.3) that, for any A-integrable function/, we have (13.16.4) f/dA= \fhdW and hence, by virtue of (13.16.3) and (13.10.3),

(13.16.5) /dA	£ ( J	\f\d\X\.

A function is said to be ^-negligible (resp. ^-measurable) if it is \|A\|-negligible (resp. \| A\| -measurable). Likewise for sets. If A and /z are two complex measures and if /is both A-integrable and /x-integrable, then / is (A + /i)-integrable, by virtue of (13.16.1) and the inequality \|A + iA S \|A\| + \|/z\| (13J.8). If A is a complex measure, a function g is said to be locally X-integrable if it is locally \|A\|-integrable. In that case, for every function /e 3ff_c(X), the function #/is A-integrable, and as in (1 3.1 3) we see that/ h-» \| gfdl is a meas- ure, denoted by g - A. If A = h - \|A\| (13.16.3), we have jfy dX =jfgh d\|A\|, or equivalently g - A = (gh) • \|A\|, so that the study of measures of the form g - A is immediately reduced to the case where A is a positive measure. In particular, it follows from (13.13.4) that (13.16.6) l<rA\| = \|<7\|-\|A\|.