Denjoy integral

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The narrow (special) Denjoy integral is a generalization of the Lebesgue integral. A function is said to be integrable in the sense of the narrow (special, ) Denjoy integral on if there exists a continuous function on such that almost everywhere, and if for any perfect set there exists a portion of on which is absolutely continuous and where where is the totality of intervals contiguous to that portion of and is the oscillation of on ; This generalization of the Lebesgue integral was introduced by A. Denjoy

who showed that his integral reproduces the function with respect to its pointwise finite derivative. The integral is equivalent to the Perron integral.

The wide (general) Denjoy integral is a generalization of the narrow Denjoy integral. A function is said to be integrable in the sense of the wide (general, ) Denjoy integral on if there exists a continuous function on such that its approximate derivative is almost everywhere equal to and if, for any perfect set , there exists a portion of on which is absolutely continuous; here Introduced independently, and almost at the same time, by Denjoy

and A.Ya. Khinchin , . The integral reproduces a continuous function with respect to its pointwise finite approximate derivative.

A totalization is a constructively defined integral for solving the problem of constructing a generalized Lebesgue integral which would permit one to treat any convergent trigonometric series as a Fourier series (with respect to this integral). Introduced by Denjoy .

A totalization differs from a totalization by the fact that the definition of the latter totalization involves an approximate rather than an ordinary limit. Denjoy  also gave a descriptive definition of a totalization . For relations between and and other integrals, see .

How to Cite This Entry:
Denjoy integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Denjoy_integral&oldid=12910
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article