Mutual existence of sum and product integrals
J.C. Helton · Pacific Journal of Mathematics · 1975
Functions are from R x R to N, where R denotes the set of real numbers and N denotes a normed complete ring.If G has bounded variation on [a, b], then I G exists if and only if X W(\ + G) exists for a ^ x < y ^ b.If each of lim x ^p+ H(p,x), Yιm x -+p-H(x,p), \im x ,y-^p+H(x, y) and \im x , y ^p H(x,y) exists, G has bounded variation on [a,b] and either G exists or Ja X W(\ + G) exists for a ^x < y ^ b, then ί HG and ί GH exist and ,Π y (l + HG) and X ΓP(1 + GH) exist for a ^ JC < y ^ b.If G has bounded variation on [a, b] and v is a nonnegative number, then I G exists and I G -I G = v if and only if x Π y (l + G) exists for α ^ x < y ^ fc and |1 + G-Π(1 + G)| = v.J. S. MacNerney [4] defines classes OA and OM of functions such that the integral-like formulas V(a,b)= Γ (W-\) and W(a, b) = β Π*(l + V)