By Stanislaw Saks

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**Additional info for Theory of the Integral**

**Example text**

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)

### Theory of the Integral by Stanislaw Saks

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