# 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 [5] also gave a descriptive definition of a totalization $(T_{2s})$. For relations between $(T_{2s})_0$ and $(T_{2s})$ and other integrals, see [6].

#### References

 [1a] A. Denjoy, "Une extension de l'intégrale de M. Lebesgue" C.R. Acad. Sci. , 154 (1912) pp. 859–862 [1b] A. Denjoy, "Calcul de la primitive de la fonction dérivée la plus générale" C.R. Acad. Sci. , 154 (1912) pp. 1075–1078 [2] A. Denjoy, "Sur la dérivation et son calcul inverse" C.R. Acad. Sci. , 162 (1916) pp. 377–380 [3] A.Ya. [A.Ya. Khinchin] Khintchine, "Sur une extension de l'integrale de M. Denjoy" C.R. Acad. Sci. , 162 (1916) pp. 287–291 [4] A.Ya. Khinchin, "On the process of Denjoy integration" Mat. Sb. , 30 (1918) pp. 543–557 (In Russian) [5] A. Denjoy, "Leçons sur le calcul des coefficients d'une série trigonométrique" , 1–4 , Gauthier-Villars (1941–1949) [6] I.A. Vinogradova, V.A. Skvortsov, "Generalized Fourier series and integrals" J. Soviet Math. , 1 (1973) pp. 677–703 Itogi Nauk. Mat. Anal. 1970 (1971) pp. 65–107 [7] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)

Just as the Lebesgue integral allows one to compute the mass corresponding to some density function, the Denjoy integral (called totalization by Denjoy also in the case 1) or 2)) allows one to compute the primitive (defined up to a constant) of some function. And, whereas for smooth functions calculating primitives is the usual way of calculating masses, in the general case the calculus of primitives (in the sense of 1) or 2)) depends on and is more involved than the calculus of masses. Denjoy gave a constructive scheme (one for $(D^*)$ and a similar one for $(D)$) to calculate when possible the totalization $F$ of a function $f$ by induction over the countable ordinal numbers, something which does not exist for similar integrals like Perron's integral: If $f$ has a totalization (for example, if $f$ is the derivative in case 1), or the approximate derivative in case 2), of some function) the construction stops at some countable ordinal number and gives $F$; if $f$ does not have a totalization, the construction never stops before $\aleph_1$. This constructive scheme uses the Lebesgue integral, and two ways of defining "improper" integrals coming from the theory of the Riemann integral for unbounded functions and due, respectively, to A.L. Cauchy and A. Harnack. For details see [7] or [a1].