On some extensions of the Perron integral on one-dimensional intervals. An approach by integral sums fulfilling a symmetry condition (Q1090785)

This paper proposes the following generalization of the Kurzweil approach to the Perron integral on an interval. A gauge being a positive function \(\delta\) on a compact interval \([a,b],\) and a Riemann-type partition \(\Delta =\{(t_ j,[x_{j-1},x_ j]):j=1,2,...,k\}\) being called \(\delta\)-fine if \[ t_ j-\delta (t_ j)\leq x_{j-1},\quad x_ j\leq t_ j+\delta (t_ j)\quad (j=1,2,...,k), \] the following concept of \(AS_{\rho}\)-integrability (asymptotically symmetric integrability) is introduced. A real function f on \([a,b]\) is \(AS_{\rho}\)-integrable over \([a,b]\) \((\rho \geq 1\) being given), if there is some real q such that for each \(\epsilon >0,\) and each \(K>0,\) there exists a gauge \(\delta\) on \([a,b]\) such that \(| q-S(f,\Delta)| <\epsilon\) for every \(\delta\)-fine \(\Delta\) such that \[ t_ 1=a,\quad t_ k=b, \] \[ t_ j-x_{j-1}<x_ j-t_ j+K(x_ j-t_ j)^{\rho}, \] \[ x_ j-t_ j<t_ j-x_{j-1}+K(t_ j-x_{j-1})^{\rho},\quad j=1,2,...,k. \] In this definition, \(S(f,\Delta)=\sum^{k}_{j=1}f(t_ j)(x_ j-x_{j- 1})\) is the usual Riemann sum. It is shown that this integral properly contains Perron's integral and a transformation theorem is proved for the \(AS_{\rho}\)-integral when \(1\leq \rho \leq 2.\) Finally, a generalization of the \(AS_{\rho}\)-integral is also proposed.