By Pier, Jean-Paul
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Extra resources for Amenable Banach algebras
Such that the difference Sôn — βδ tends to zero, or, what amounts to the same thing, σδη fias a definite limit for any choice of points ξ^\ This limit A in fact gives the value of the integral. Now, sô -> A and Sôn -*■ A. This theorem follows directly from , the sub-intervals Δ^ having no common points in the present case. I t should be mentioned t h a t t h e sequence of subdivisions δη mentioned in the theorem need not necessarily be a sequence with the sub-intervals becoming indefinitely smaller.
The integral defined in  will be called simply the Stieltjes integral, or the original Stieltjes integral, in order to distinguish it from the general integral. We now give the conditions for the existence and the properties of the general integral. THEOREM 1. The necessary and sufficient condition for the existence of integral (101) is that there exist a sequence of subdivisions δη (n = 1, 2, . . ) such that the difference Sôn — βδ tends to zero, or, what amounts to the same thing, σδη fias a definite limit for any choice of points ξ^\ This limit A in fact gives the value of the integral.
We put h (x) = sup h (xk). Xk
Amenable Banach algebras by Pier, Jean-Paul