ON A THEOREM OF Г. RIESZ

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Then the inequality (*) subsists with c = 1, for any p, l<p<oo. theless 9({^i})=l, although ya({a^i}) = 0.

2 . 7 . Remark 2 (to Theorem I). A special consequence of the condition (*) of Theorem I is that \(р{А)\йс^'^^i{Af-'^^'P for every Ae%t Even if 3Ï together with О is a field ( = ^), this special inequality is, however, not a sufficient condition. As an example, let X be the interval 0 < a; < 1 ; let ^ be all finite unions of intervals of the form a^x<b contained in Jl, together with the empty set; let /л be the Borel measure in X; and let

q ) { A ) = \ x-^^^dx

A

for every A from the field fÇ. Since x-'^^P is a decreasing positive tion, we have

0 ^ (p(A) й [ x-^^Pdx = [р/{р-1)ЫАУ~^^^ •

0

Nevertheless , the derivative x-^^^ of cp with respect to /u is not in the class

3 . The extremie cases jp = cx> and jp = l.

3 . 1 . The case p = oo. Under the same assumptions as for the case l<p<oo (see Section 2.1), except that we now, like in Remark 1, admit sets of measure zero in %, the following result may be obtained :

Theorem II. In order that there exist a bounded, ^-measurable function f{x), defined in X, with the property that

( p ( A ) = \f(x) fi(dX) for every Ае'й ,

A

it is necessary and sufficient that there is a finite constant с so that the

inequality

ЫА ) \ й С(г{А)

holds for every set ^4 g 21. — The function f is then essentially uniquely determined, and the smallest possible value of с is ess ^'^'Çxex\Î(^)\'

A proof of this theorem may be obtained from the proof of Theorem I in the version described in Remark 1 (see 2.6), by obvious modifications.

3 . 2 . The case jp = l. This second limiting case is essentially different from the case l<p<oo. Again, we do not assume that 1л(А)>0 for