On some variational measures related to the wide Denjoy integral and its counterparts

Abstract

Weak and q-weak variationalmeasures defined by Brian S. Thomson [On VBG functions and the Denjoy–Khintchine integral, Real Analysis Exchange, 41(1) (2015/16), 173–226] are shown to coincide with variational measures resulting from Riemann definitions of some wide Denjoy type integrals. This fact is applied in characterizations of these integrals, via absolute continuity of weak and q-weak measures. In related results, it is discussed if these Riemann definitions can be essentially simplified. The paper is a follow-up to a paper by the author published some time ago in the Rendiconti [On Riemann-type definition for the wide Denjoy integral, Rendiconti del Seminario Matematico della Università di Padova, 126 (2011), 175–200].