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116 XIII INTEGRATION
increasing sequence (gn) of functions belonging to J such that for each n
we have /„ £ gn and ju*(#M) g /**(/„) -f £. Then, if we put /= sup/n and |
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g = sup gn , we shall have geJ,f^g, n*(g) = sup ju*(^n) ^ sup /**(/„) + e
n n
by virtue of (13.5.2), and finally fi*(f) ^ M*(#) ^ SUP /**(/«) + e- Since s was
R
arbitrary, this will complete the proof.
By definition, for each n there exists hneJ such that /„ <g An and
Ai*(/J ^ M*(*») ^ /**(/») + 2~"8. Put ^ = supC/fi, . . . 5 hn\ so that ^ e ^ (12.7.5). Clearly the sequence (gn) is increasing, and/n ^ gn . We shall show by induction on n that
(13.5.7.1 ) n*(ffn) ^ A**CA) + e(l - 2-").
This is evident from the definitions when n = 1. Suppose that (13.5.7.1) is
true, and remark that gn+1 = sup(hn+1, gn) and/,, ^ inf(/zn+1, ^rt). Since the functions hn+1 and gn do not take the value — oo, we have
sup(ftn+ !, flfB) + inf(/in+ 1? 0n) = /zn+ 1 + ^
and therefore, by virtue of (13.5.3) and the fact that n*(0n) and ju*(/zrt+1) are
finite, |
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/**(**+ i)-
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by the inductive hypothesis (1 3.5.7.1 ). Q.E.D.
It should be noted that for a decreasing sequence (/„), it is not necessarily
true that ju*/mf/n\ = inf /**(/„), even if all the numbers n*(fn) are finite
(Section 13.8, Problem 13).
(13.5.8) If(fn) is any sequence of functions ^0, then
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(00 \ 00
Z/.U EM*(/B).
n~l / n=l |
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