Burkill integral

A concept which was introduced by J.C. Burkill

to determine surface areas. The Burkill integral is introduced in its modern form for the integration of a non-additive function $F(J)$ over an $n$-dimensional segment (a block). Let $R$ be a set that can be represented as a sum (union) of a finite number of segments (such a set is called a figure). Each representation $R=\bigcup J_k$ is called a subdivision of the figure $R$. The upper and the lower Burkill integral of the segment function $F(J)$ over the figure $R$ are, respectively, the upper and lower limits of the sums $\sum_kF(J_k)$ for all possible subdivisions as the maximum of the diameters of the segments involved in the subdivision tends to zero. If these integrals are equal, their common value is the Burkill integral of $F$ over $R$ and is denoted by $\int_RF$. If $F$ is integrable over $R$, then $F$ is integrable on each figure $R_1\subset R$. This enables one to introduce an indefinite Burkill integral, which is an additive set function. If $F$ is continuous, the indefinite Burkill integral is continuous as well.

The concept of the Burkill integral can be generalized to include the case of a set function defined on some class of subsets of an abstract measure space. This class must meet a number of requirements; in particular, each set of the class must permit a subdivision into sets also of this class that have a measure as small as one pleases. The Burkill integral can then be defined for any set in the class in analogy with the $n$-dimensional case, the respective limits being taken as the maximum of the measures of the constituent sets tends to zero. The Burkill integral can be naturally generalized to set functions with values in a commutative topological group. The Burkill integral is less general than the subsequently introduced Kolmogorov integral, which is also known as the Burkill–Kolmogorov integral. Any function that is Burkill-integrable is also Kolmogorov-integrable after a suitable ordering of the subdivisions. The converse statement is true only if certain additional conditions are satisfied. The Burkill integral is used in constructing the Denjoy integral in different spaces.

The name of Burkill integral is also given to a number of generalizations of the Perron integral ($AP$-, $CP$-, $SCP$-integrals), which were also introduced by Burkill. Instead of ordinary derivatives certain generalized derivatives are used in the definition of these integrals. These Burkill integrals are used in the theory of trigonometric series.

References

[1a]	J.C. Burkill, "Functions of intervals" Proc. London Math. Soc. (2) , 22 (1924) pp. 275–310
[1b]	J.C. Burkill, "The expression of area as an integral" Proc. London Math. Soc. (2) , 22 (1924) pp. 311–336
[2]	S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)
[3]	P.I. Romanovskii, "l'Intégrale de Denjoy en espaces abstractes" Mat. Sb. , 9 (51) : 1 (1941) pp. 67–120
[4]	J.C. Burkill, "Integrals and trigonometric series" Proc. London Math. Soc. (3) , 1 (1951) pp. 46–57