Difference between revisions of "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 | + | {{TEX|done}} |
| + | 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==== | ====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> | <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> | ||
Latest revision as of 18:13, 4 November 2014
A general scheme for constructing an integral, including the Lebesgue–Stieltjes integral, the Burkill integral, the Hellinger integral, etc. Introduced by A.N. Kolmogorov [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
| [1] | A. [A.N. Kolmogorov] Kolmogoroff, "Untersuchungen über den Integralbegriff" Math. Ann. , 103 (1930) pp. 654–696 |
Kolmogorov integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kolmogorov_integral&oldid=11583