Topics in Banach Space Integration

Equiintegrability and controlled convergence for the Henstock-Kurzweil integral

“…By Theorem 3.3.7, [7], we get also that each F n is the primitive of HK-integrable function f (E) n . Therefore, we obtain by Theorem 8, [6] …”

“…and consequently the sequence (h ( * ) n k (s)) converges to f (s) for each s ∈ E. Hence, by Theorem 3.3.4 and Theorem 3.3.7 in [7], for each I ∈ I, we have…”

“…Let z be an arbitrary element of Z. Since (f n ) is HK-equiintegrable, by Lemma 3.5.6, [7], for the given > 0 there exists a gauge δ 1 on Z such that for every δ 1 -fine Z-tagged HK-partition π in [0, 1], we have…”

“…Hence, we obtain by Theorem 3.3.7 in [7] that the function f is HK-integrable on [0, 1] and lim n→∞ (HK) [0,1] …”

Convergence theorems for the Birkhoff integral

“…A classical exposition of this theory can be found in [5] and [3]; see also the recent monograph [14] including the McShane and Kurzweil-Henstock integrals. The Birkhoff integral for Banach space valued functions, located strictly between the Bochner and Pettis integrals, was introduced in 1935 (see [1]).…”