By Curtiss D.R.

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So Theorem1 reaffirms the result . 3. 5 Upper and Lower Integrals Boolean Algebra of Figures. as Functions on the The next result shows that lower and upper integrals are additive on finite unions of nonoverlapping figures where the indeterminate form cx~ - o¢ does not occur. But even for the indeterminate case there are appropriate conclusions for finite summants -cx~ < S < ~. THEOREM 1. Let S be a summant on a union C of nonoverlapping figures A, B. Then the lower and upper integrals are additive in the following sense: If the right-hand side is indeterminate then fcS = -~x~.

LEMMA B. Let the summantsT~ >_ 0 on K for i = O, 1, 2,... such that given 0 < c < 1 there exists an integer-valued function n(t) >_ 1 on K satisfying (1) ~(~) cTo(I,t) <_ ~-~Ti(l,t) i----1 for every tagged cell (I, t) in K. -. by letting Si(I,t) - Ti(I,t) for i <_ n(t),O for i > n(t). Then cTo <_ ~i~=1 Si by (1). So for every gauge a on (3) c To <_ cT(oa)(K) <_ ~ S~)(K). i=l Let e > 0begiven. Since0_< Si_< Ti for alli >_ lwecan choose for each such i a gauge ~i on K small enough so that (4) ( )_< for i ---- 1,2,--..

In (a, b) such gha~ r~ ~ By the monotone convergence in (1) S (2) Nowsincetheupperintegral is additive overabutting ceils, (a) s= s+ s 4--I By (2) and (3) we get the convergent series (4) S + i=1 ’~--1 Nowfor all n _> 1, (5) l[a,b) _< l[a,~) + ~ l(ri_l,ri+l). 9 INTEGRATION ~OVERARBITRARY INTERVALS 45 Applying Theorem 1 to (5) and invoking (2) we S. _

### Analytic functions of a complex variable by Curtiss D.R.

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