By Edgar Asplund; Lutz Bungart

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Post yr be aware: First released March 2d 1979

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1111111 Proof. For each n pick an so that Ixp y > a,- We can assume that a1 < a2 Problem 10. Every convergent net is a Cauchy net. 1111111 Problem 11. If A is an index set and {x, :a E A} is summable, and A = U A, where the A, are disjoint subsets of A, then {x,: a E A,} is summable for each n and Problem 12. (i) If {G,} is summable over N x N,then 2 THE RIEMANN INTEGRAL AS A UMlT OF SUMS 19 (ii) Suppose the iterated (ordered) sums both exist. Does it follow that is summable? (iii) Suppose the iterated sums both exist and h, 2 0 for all m, n.

What we must show is m [(El u E2) n TI + m [(El u Ed’ n = m(T) ; or, in terms of Fig. 1, Cutting the test set % U Similarly, cutting 5U Cutting T with El gives with the measurable set E2 gives with E2 gives 34 A PRIMER OF LEBESGUE INTEGRATION Combining (7),(8), (9)we can write Now cut Ti U T2 U with El and then use (7): From (11)and (10)we have the desired equality Corollary. Finite unions and finite intersections of measurable sets are measurable. El - E2 is measurable i f El, E2 are. Proof. Notice that E satisfies the characterizing equation m(E n T ) + m(E’ n T ) = m(T) for all T if and only if E’ does.

Let P @ Q mean that )I QII 5 11 P 11. Show that @ makes the partitions P and the pairs ( P , c) into directed sets. Let lim stand ll1’llk+O for the limit with the direction 0. Use Problem 14 to show that if lim R( f, P , c) = I, then f is Riemann integrable with /I p lI+O integral I . Conversely, lim R( f, P , c) = I implies lim R( f, P , c) = 1. P I1 P II+o Prove this. Hint: Let Po = {xo,. . , x,} be a partition of [a, b] such that U( f , PO)- L( f, Po) < E , and so J R (f, Po, co) - I I < E .