Download PDF by Stanislaw Saks: Theory of the Integral

By Stanislaw Saks

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E Lebesguc'H Theorem on term by term integration. {f,,(x)} fulfilling, \1,,(x)\ f g, (x) du (x) E ' K a sequence of functions measurable (9E) on a set E, a for function s(x) integrable (9E) on JS, the inequality be <: s(x) for n=l, Then ... 2, fn d > lim inf f n dp. 12) lim sup f fn " dfi ^ sequence {f,,} converges on E term by term, i. e. 13) f lim sup / du. E lim ffn dn to a function f, the sequence CHAPTER 30 Proof. Let The I. g(x) integral in = lim inf fn (x) an abstract space. and let n We may clearly suppose from Fatou's lim inf f " Lemma f n )dp^* (s [ We throughout E.

2Fn , which is a set ($). 3) 43 Continuous and semi -continuous functions. [{3] Theorem. For a set A f(x) > a] semi-continuous on a each number a, the e ; X be closed in A, e. i. 4) function of a point f(x) [E[xcA; X expressible as the common part of a]] A with a set (5). We Proof. need only consider the case of upper semi-continuous functions, as the other case follows by change of sign. 4). bitrary number, and x For each r>0, the sphere S(# r) then contains points of that set, and this requires M^(/; xQ r )^ a an(i 80 M^(/; x Q )^a.

And let n We may clearly suppose from Fatou's lim inf f " Lemma f n )dp^* (s [ We throughout E. lim inf E (* + /n) (* then derive + g)dp and B which gives at once the h)dp, (s = lim sup /(#). 12). 13). 13. Absolutely continuous additive functions of a set. The fact that the indefinite integral of a function integrable ($, p) on a set E is, on E, an additive function of a set (SE), raises the problem of characterizing directly the additive functions expressible as indefinite integrals. iable, If we restrict ourselves to we may regard indefinite the Lebesgue integral of functions of a real varintegrals as functions of an interval, or, what comes to the same thing, as functions of a real variable.

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Theory of the Integral by Stanislaw Saks

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