Approximate limit

A limit of a function as over a set for which is a density point. In the simplest case is a real-valued function of the points of an -dimensional Euclidean space; in the more general case it is a vector function. The approximate limit is denoted by

In general, the existence of an ordinary limit does not follow from the existence of an approximate limit. An approximate limit displays the elementary properties of limits — uniqueness, and theorems on the limit of a sum, difference, product and quotient of two functions.

Let be a density point of the domain of definition of a real-valued function . If the ordinary limit exists, the approximate limit also exists and is equal to it. The approximate upper limit of a function at a point is the lower bound of the set of numbers (including ) for which is a point of dispersion of the set . Similarly, the approximate lower limit of a function at a point is the upper bound of the set of points (including ) for which is a point of dispersion of the set . These approximate limits are denoted, respectively, by

An approximate limit exists if and only if the approximate upper and lower limits are equal; their common value is equal to the approximate limit.

If is real, one-sided (right and left) approximate upper and lower limits are also used ( must then be, respectively, a right-hand or left-hand density point in the domain of definition of the function). For the approximate right upper limit the following notation is used:

with corresponding notations for the other cases. If the approximate right upper and lower limits coincide, one obtains the right approximate limit; if the approximate left upper and lower limits coincide, one obtains the left approximate limit.

Approximate limits were first utilized by A. Denjoy and A.Ya. Khinchin in the study of the differential connections between an indefinite integral (in the sense of Lebesgue and in the sense of Denjoy–Khinchin) and the integrand (cf. Approximate continuity; Approximate derivative).

References

[1]	S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)

Comments

A point of dispersion is defined as follows: Let be a set in . Let be the completely-additive set function defined for measurable by , the outer measure of . Let be any point in . The upper strong derivative and lower strong derivative and are called, respectively, the upper outer density and lower outer density of at . The point is a point of density for a set if the outer density of at is 1 and it is a point of dispersion if the outer density of at is zero. If is measurable, almost-all points of are points of density and almost-all points of its complement are points of dispersion. The latter condition is also sufficient for to be measurable.

References

[a1]	A.M. Bruckner, "Differentiation of real functions" , Springer (1978)
[a2]	M.E. Munroe, "Introduction to measure and integration" , Addison-Wesley (1953) pp. 111
[a3]	B.S. Thomson, "Real functions" , Springer (1985)