Darboux sum

A sum of special type. Let a real function be defined and bounded on a segment , let be a decomposition of :

and set

The sums

are known, respectively, as the lower and upper Darboux sums. For any two decompositions and of the inequality is valid, i.e. any lower Darboux sum is no larger than an upper. If

is a Riemann sum, then

The geometric meaning of the lower and upper Darboux sums is that they are equal to the planar areas of stepped figures consisting of rectangles whose base widths are and with respective heights and (see Fig.) if . These figures approximate, from the inside and outside, the curvilinear trapezium formed by the graph of , the abscissa axis and the rectilinear segments and (which may degenerate into points).

Figure: d030160a

The numbers

(1)

are called, respectively, the lower and the upper Darboux integrals of . They are the limits of the lower and the upper Darboux sums:

where

is the fineness (mesh) of the decomposition . The condition

(2)

is necessary and sufficient for a function to be Riemann integrable on the segment . Here, if condition (2) is met, the value of the lower and the upper Darboux integrals becomes identical with the Riemann integral

With the aid of Darboux sums, condition (2) may be formulated in the following equivalent form: For each there exists a decomposition such that

The condition

is also necessary and sufficient for the Riemann integrability of on . Here

where is the oscillation (cf. Oscillation of a function) of on

The concept of lower and upper Darboux sums may be generalized to the case of functions of several variables which are measurable in the sense of some positive measure . Let be a measurable (for example, Jordan or Lebesgue) subset of the -dimensional space, and suppose is finite. Let be a decomposition of , i.e. a system of measurable subsets of such that

(3)

(4)

Let a function be bounded on and let

(5)

The sums

(6)

are also said to be, respectively, lower and upper Darboux sums. The lower and the upper integrals are defined by formulas (1). For Jordan measure, their equality is a sufficient and necessary condition for the function to be Riemann integrable, and their common value coincides with the Riemann integral. For Lebesgue measure, on the other hand, the equality

is always valid for bounded Lebesgue-measurable functions.

In general, if is a complete -additive bounded measure, defined on a -algebra , if is a bounded measurable real-valued function on , if is a decomposition of a set into -measurable sets which satisfy the conditions (3) and (4), and if the Darboux sums and are defined by formulas (5) and (6), while the integrals and are defined by the formulas (1), in which is always understood to mean the measure under consideration, then

A generalization of the Darboux sums to unbounded -measurable functions defined on sets are the series (if they are absolutely convergent)

(7)

where is a decomposition of (this decomposition consists, generally speaking, of an infinite number of -measurable sets which satisfy condition (4) and are, of course, such that ), while and are defined by (5). In (7) (as in (6) above) it is assumed that . If and are again defined according to (1) and and are now defined in the sense of (7) and exist for each , then . If the value is finite, then is integrable with respect to and .

Named after G. Darboux [1].

References

[1]	G. Darboux, Ann. Sci. Ecole Norm. Sup. Ser. 2 , 4 (1875) pp. 57–112
[2]	V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)
[3]	L.D. Kudryavtsev, "Mathematical analysis" , 1–2 , Moscow (1973) (In Russian)
[4]	S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)