ABSTRACT
Treating approximating sums of the classical form (2) E. J. McShane [34] used gauge direction but allowed the tags to be any points in K. So (/, t) is 5-fine for every cell I in K which lies in the 5-neighborhood (t — 5 (i),t + 5(<)) of t whether I con tains t or not. McShane’s integral is equivalent to the Lebesgue integral on K but has the advantage of avoiding the tedious preliminaries required by the usual approaches to the Lebesgue integral. But, sharing the narrow scope of the Lebesgue in tegral, McShane’s integral precludes conditional integrability. This is a serious disadvantage. End point tagging is essential for a unified approach to integration on R.
