Darboux sum
A sum of special type. Let a real function be defined and bounded on a segment
, let
be a decomposition of
:
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and set
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The sums
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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
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is a Riemann sum, then
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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:
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where
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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
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With the aid of Darboux sums, condition (2) may be formulated in the following equivalent form: For each there exists a decomposition
such that
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The condition
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is also necessary and sufficient for the Riemann integrability of on
. Here
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where is the oscillation (cf. Oscillation of a function) of
on
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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
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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
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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) |
Darboux sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_sum&oldid=46580