# Denjoy integral

The narrow (special) Denjoy integral is a generalization of the Lebesgue integral. A function $f$ is said to be integrable in the sense of the narrow (special, $D^*$) Denjoy integral on $[a,b]$ if there exists a continuous function $F$ on $[a,b]$ such that $F'=f$ almost everywhere, and if for any perfect set $P$ there exists a portion of $P$ on which $F$ is absolutely continuous and where

$$\sum_n\omega(F;(\alpha_n,\beta_n))<\infty,$$

where $\{(\alpha_n,\beta_n)\}$ is the totality of intervals contiguous to that portion of $P$ and $\omega(F;(\alpha,\beta))$ is the oscillation of $F$ on $(\alpha,\beta)$;

$$(D^*)\int\limits_a^bf(x)dx=F(b)-F(a).$$

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 $D^*$ integral is equivalent to the Perron integral.

The wide (general) Denjoy integral is a generalization of the narrow Denjoy integral. A function $f$ is said to be integrable in the sense of the wide (general, $D$) Denjoy integral on $[a,b]$ if there exists a continuous function $F$ on $[a,b]$ such that its approximate derivative is almost everywhere equal to $f$ and if, for any perfect set $P$, there exists a portion of $P$ on which $F$ is absolutely continuous; here

$$(D)\int\limits_a^bf(x)dx=F(b)-F(a).$$

Introduced independently, and almost at the same time, by Denjoy

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

A totalization $(T_{2s})_0$ 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 $(T_{2s})$ differs from a totalization $(T_{2s})_0$ 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 $(T_{2s})$. For relations between $(T_{2s})_0$ and $(T_{2s})$ and other integrals, see .

How to Cite This Entry:
Denjoy integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Denjoy_integral&oldid=32641
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article