View source for Kolmogorov integral

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A general scheme for constructing an integral, including the [[Lebesgue–Stieltjes integral|Lebesgue–Stieltjes integral]], the [[Burkill integral|Burkill integral]], the [[Hellinger integral|Hellinger integral]], etc. Introduced by A.N. Kolmogorov [[#References|[1]]]. One considers a directed family of partitions of a space $E$ of arbitrary nature. A set function $\Phi$ (generally many-valued) is defined on the elements of the partition. The sum of the values of this function taken over all elements of the partition gives a many-valued function of the partition. In particular, this sum is a generalization of the Riemann sum $\sum_if(\xi_i)\Delta x_i$ where the multi-valuedness is a consequence of the arbitrariness in the choice of the points $\xi_i$ on the elements of the partition. The directed limit of the partition functions then defines the Kolmogorov integral $\int_Ed\Phi$. The Kolmogorov integral is considered both for finite and countable partitions. The Kolmogorov integral can be considered for functions with values in a commutative topological group.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. [A.N. Kolmogorov] Kolmogoroff,   "Untersuchungen über den Integralbegriff"  ''Math. Ann.'' , '''103'''  (1930)  pp. 654–696</TD></TR></table>