Henstock-Kurzweil Integration: Its Relation to Topological by Jaroslav Kurzweil

By Jaroslav Kurzweil

Henstock-Kurzweil (HK) integration, that's in accordance with fundamental sums, may be got via an not noticeable switch within the definition of Riemann integration. it's an extension of Lebesgue integration and there exists an HK-integrable functionality f such that its absolute worth |f| isn't HK-integrable. during this textual content HK integration is handled basically on compact one-dimensional periods. the concept that of convergent sequences is transferred to the set P of primitives of HK-integrable features; those convergent sequences of services from P are referred to as E-convergent. the most effects are: there exists a topology U on P such that (1) (P,U) is a topological vector house, (2) (P,U) is whole, and (3) each E-convergent series is convergent in (P,U). however, there isn't any topology U pleasing (2),(3) and (P,U) being a in the neighborhood convex house.

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Put 3 - x = {Z - x\ Z G 3} so that 3 - x -* 0 in X and 3 - x is a Cauchy filter in ( P , W). 7. (p) C P . Theorem. Let U be a topology on P Then (i) W is tolerant to Q-convergence if and only if + (ii) for x e U eU there exists p : D - R such that x + R(p) C U. 4. 8. For Z C X denote the convex hull of Z by convZ. HENSTOCK-KURZWEIL 52 INTEGRATION Definition. Let Wlc = W*LC(P,Q) = {convR(p);p: D - M+}. Let U1C be the set of U c P with the property if y G £/, then there exists W G 2H/,c s u c n * n a t 2/ + W U.

Let now j,A: e N and let A = {(t, J)} G S(I,I,6(j,-) partition of 7. be a 46 HENSTOCK-KURZWEIL INTEGRATION Put A 2 = {(*, J) 6 A; t e M*}, Ai = A \ A 2 , 7i(fc) = *i(fc)(-0 for A;€ N. Then (cf. /Il < J Z l*i(fc)(J) - 9i(k)(t)\J\\ < A £ A \FHk)(J) - 9m(t)\J\\ + ^ \FiW(J)\ < A, < 2~ J '- 2 + 2 _ J ' - 2 + ^ I ' l < 2~j~2 + 2~J~2 + 2~j~1 < 2~j. 12 is complete. - ^ F. The proof of □ The next theorem shows that ^-convergence and <5-convergence are equivalent from the point of view of topology. 13 Definition.

11 Lemma. Let ■d, ( : I —> [0,1], iet £ be measurable and ((t) > fl(t) almost everywhere. e. Proof. Put M = {t G 7;C(0 < #(<)}• By assumption M G A/". For a,(3eR+ put if = H(a, (3) = {te /; C(0 < a, fo{t) > a + 20}. ii) \{C(t) < Mt)}\ > o, then there exist a,0 such that \H\ > 0. 77 is measurable. Let t e H D densH. ). Since i G if, we have £ G cless£ so that £ G clessi? D densii. 7 implies that there exists z G E n i i \ M. (z) because z G I\M. 11) cannot hold and the proof is complete. □ 2. 12.

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