## On some extensions of the Perron integral on one-dimensional intervals. An approach by integral sums fulfilling a symmetry condition.(English)Zbl 0622.26007

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.
Reviewer: J.Mawhin

### MSC:

 26A39 Denjoy and Perron integrals, other special integrals