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15 THE LEBESGUE-NIKODYM THEOREM 191

This shows first of all that iff is /z-negligible then v(/) = 0. Passing to the
quotient by the subspace Ji of /^-negligible functions, the linear form v on
jjf _R(X) therefore defines a linear form /»-» v(/) on the subspace «#* _R(X) of
L_R(X, u) which is the canonical image of ^T_R(X) in L«(X, /*). The inequality
(13.5.1.1) shows that this linear form is continuous (5.5.1) on the subspace
jf _R(X); since the latter is dense in Lg(X, u) (13.11.6), the linear form/W v(/)
extends by continuity to a linear form on the Hilbert space L_R(X, ^) (5.5.4).
Hence there exists a function g e J2?_R(X, /j) such that v(/) = X#/) for all
/e jf_H(X) (6.3.2). Since # is locally //-integrable (13.1 3), it follows that v =g • fi.

(II) General case. There exists a partition of X into a sequence of com-
pact sets K_n and a /^-negligible set N (13.9.2). If we put M = X - N = (J K_n ,

then (p_M is locally /j-integrable and 1 — <p_M is /^-negligible, hence (13.14.4)
<p_M • in = /z. Put /z_n = <p_Kn • /z and v_n = <p_Kn • v. Since <p_Kn is the lower envelope
of a decreasing sequence of functions ^0 belonging to «^*_R(X), the relation
v ^ fi implies v_n ^ p,_n for all n; furthermore, v_n is bounded (13.14.4). Hence,
by (I) above, there exists a locally /vintegrable function g_n such that
*»=&,-/*„ = (0_n<PK_n) ' M = \ff_n<PK_n\ ' V ((13.13.5) and (13.14.5)), and the func-
tion g_n <p_Kn is locally /i-integrable. Let g denote the function which is equal
to \g_n9&_n\ °ⁿ ^each K_n and is zero on N. Then g is the sum of the series

Z \0n⁽PK_n\- If /is any function ^0 in Jf_R(X), and m is any integer, then

f/w \ m r m r

\f ZI0-PKJ UA*=S /I^^KJ^=Z
J \n=l / n=l J n=l J

_Kndv£ f

and therefore ((13.8.4) and (13.13.1)) the function # is locally //-integrable.

Also, since /= £/%„ almost everywhere (with respect to ju), we have

n- 1

(13.8.4) \fg d/j, = I fcp_M dv. To show that v = g • /j, it remains to be proved
that N is also ^-negligible. But by definition (1 3.5.5), for each e > 0 there exists
a function h e ,/ such that (p^^h and X^) ^ ^fi- Since // is the upper envelope
of an increasing sequence of functions ^0 belonging to J>T_R(X), the inequality
v <* /z implies that v*(/z) ^ /z(/z) ^ e, and the proof is complete.

The following lemma will be generalized later (13.15.8):

(13.15.2) If(^j)i^j<,_r is any finite sequence of complex measures on X, there
exists a positive measure 1 such that each /Zy is a measure with base L



	15 THE LEBESGUE-NIKODYM THEOREM 191 This shows first of all that iff is /z-negligible then v(/) = 0. Passing to the quotient by the subspace Ji of /^-negligible functions, the linear form v on jjf _R(X) therefore defines a linear form /»-» v(/) on the subspace «#* _R(X) of L_R(X, u) which is the canonical image of ^T_R(X) in L«(X, /). The inequality (13.5.1.1) shows that this linear form is continuous* (5.5.1) on the subspace jf _R(X); since the latter is dense in Lg(X, u) (13.11.6), the linear form/W v(/) extends by continuity to a linear form on the Hilbert space L_R(X, ^) (5.5.4). Hence there exists a function g e J2?_R(X, /j) such that v(/) = X#/) for all /e jf_H(X) (6.3.2). Since # is locally //-integrable (13.1 3), it follows that v =g • fi. (II) General case. There exists a partition of X into a sequence of com- pact sets K_n and a /^-negligible set N (13.9.2). If we put M = X - N = (J K_n , n then (p_M is locally /j-integrable and 1 — <p_M is /^-negligible, hence (13.14.4) <p_M • in = /z. Put /z_n = <p_Kn • /z and v_n = <p_Kn • v. Since <p_Kn is the lower envelope of a decreasing sequence of functions ^0 belonging to «^_R(X), the relation v ^ fi* implies v_n ^ p,_n for all n; furthermore, v_n is bounded (13.14.4). Hence, by (I) above, there exists a locally /vintegrable function g_n such that »=&,-/„ = (0_n<PK_n) ' M = \ff_n<PK_n\ ' V ((13.13.5) and (13.14.5)), and the func- tion g_n <p_Kn is locally /i-integrable. Let g denote the function which is equal to \g_n9&_n\ °ⁿ ^each K_n and is zero on N. Then g is the sum of the series 00 Z \0n⁽PK_n\- If /is any function ^0 in Jf_R(X), and m is any integer, then

	f/w \ m r m r \f ZI0-PKJ UA=S /I^^KJ^=Z J \n=l / n=l J n=l J _Kndv£ f*

	and therefore ((13.8.4) and (13.13.1)) the function # is locally //-integrable. 00 Also, since /= £/%„ almost everywhere (with respect to ju), we have n- 1 (13.8.4) \fg d/j, = I fcp_M dv. To show that v = g • /j, it remains to be proved that N is also ^-negligible. But by definition (1 3.5.5), for each e > 0 there exists a function h e ,/ such that (p^^h and X^) ^ ^fi- Since // is the upper envelope of an increasing sequence of functions ^0 belonging to J>T_R(X), the inequality v <* /z implies that v*(/z) ^ /z(/z) ^ e, and the proof is complete.

	The following lemma will be generalized later (13.15.8):

	(13.15.2) If(^j)i^j<,_r is any finite sequence of complex measures on X, there exists a positive measure 1 such that each /Zy is a measure with base L