Darboux integral

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In real analysis, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal.[1] The definition of the Darboux integral has the advantage of being easier to apply in computations or proofs than that of the Riemann integral. Consequently, introductory textbooks on calculus and real analysis often develop Riemann integration using the Darboux integral, rather than the true Riemann integral.[2] Moreover, the definition is readily extended to defining Riemann–Stieltjes integration.[3] Darboux integrals are named after their inventor, Gaston Darboux (1842–1917).

Definition

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The definition of the Darboux integral considers upper and lower (Darboux) integrals, which exist for any bounded real-valued function   on the interval   The Darboux integral exists if and only if the upper and lower integrals are equal. The upper and lower integrals are in turn the infimum and supremum, respectively, of upper and lower (Darboux) sums which over- and underestimate, respectively, the "area under the curve." In particular, for a given partition of the interval of integration, the upper and lower sums add together the areas of rectangular slices whose heights are the supremum and infimum, respectively, of f in each subinterval of the partition. These ideas are made precise below:

Darboux sums

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Lower (green) and upper (green plus lavender) Darboux sums for four subintervals

A partition of an interval   is a finite sequence of values   such that

 

Each interval   is called a subinterval of the partition. Let   be a bounded function, and let

 

be a partition of  . Let

 


The upper Darboux sum of   with respect to   is

 

The lower Darboux sum of   with respect to   is

 

The lower and upper Darboux sums are often called the lower and upper sums.

Darboux integrals

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The upper Darboux integral of f is

 

The lower Darboux integral of f is

 

In some literature, an integral symbol with an underline and overline represent the lower and upper Darboux integrals respectively:

 

and like Darboux sums they are sometimes simply called the lower and upper integrals.

If Uf = Lf, then we call the common value the Darboux integral.[4] We also say that f is Darboux-integrable or simply integrable and set

 

An equivalent and sometimes useful criterion for the integrability of f is to show that for every ε > 0 there exists a partition Pε of [a, b] such that[5]

 

Properties

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  • For any given partition, the upper Darboux sum is always greater than or equal to the lower Darboux sum. Furthermore, the lower Darboux sum is bounded below by the rectangle of width (ba) and height inf(f) taken over [a, b]. Likewise, the upper sum is bounded above by the rectangle of width (ba) and height sup(f).
     
  • The lower and upper Darboux integrals satisfy
     
  • Given any c in (a, b)
     
  • The lower and upper Darboux integrals are not necessarily linear. Suppose that g:[a, b] → R is also a bounded function, then the upper and lower integrals satisfy the following inequalities:
     
  • For a constant c ≥ 0 we have
     
  • For a constant c ≤ 0 we have
     
  • Consider the function
     
then F is Lipschitz continuous. An identical result holds if F is defined using an upper Darboux integral.

Examples

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A Darboux-integrable function

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Suppose we want to show that the function   is Darboux-integrable on the interval   and determine its value. To do this we partition   into   equally sized subintervals each of length  . We denote a partition of   equally sized subintervals as  .

Now since   is strictly increasing on  , the infimum on any particular subinterval is given by its starting point. Likewise the supremum on any particular subinterval is given by its end point. The starting point of the  -th subinterval in   is   and the end point is  . Thus the lower Darboux sum on a partition   is given by

 

similarly, the upper Darboux sum is given by

 

Since

 

Thus for given any  , we have that any partition   with   satisfies

 

which shows that   is Darboux integrable. To find the value of the integral note that

 
Darboux sums
Darboux upper sums of the function y = x2
Darboux lower sums of the function y = x2

A nonintegrable function

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Suppose we have the Dirichlet function   defined as

 

Since the rational and irrational numbers are both dense subsets of  , it follows that   takes on the value of 0 and 1 on every subinterval of any partition. Thus for any partition   we have

 

from which we can see that the lower and upper Darboux integrals are unequal.

Refinement of a partition and relation to Riemann integration

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When passing to a refinement, the lower sum increases and the upper sum decreases.

A refinement of the partition   is a partition   such that for all i = 0, …, n there is an integer r(i) such that

 

In other words, to make a refinement, cut the subintervals into smaller pieces and do not remove any existing cuts.

If   is a refinement of   then

 

and

 

If P1, P2 are two partitions of the same interval (one need not be a refinement of the other), then

 

and it follows that

 

Riemann sums always lie between the corresponding lower and upper Darboux sums. Formally, if   and   together make a tagged partition

 

(as in the definition of the Riemann integral), and if the Riemann sum of   is equal to R corresponding to P and T, then

 

From the previous fact, Riemann integrals are at least as strong as Darboux integrals: if the Darboux integral exists, then the upper and lower Darboux sums corresponding to a sufficiently fine partition will be close to the value of the integral, so any Riemann sum over the same partition will also be close to the value of the integral. There is (see below) a tagged partition that comes arbitrarily close to the value of the upper Darboux integral or lower Darboux integral, and consequently, if the Riemann integral exists, then the Darboux integral must exist as well.

See also

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Notes

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  1. ^ David J. Foulis; Mustafa A. Munem (1989). After Calculus: Analysis. Dellen Publishing Company. p. 396. ISBN 978-0-02-339130-9.
  2. ^ Spivak, M. (1994). Calculus (3rd. edition). Houston, TX: Publish Or Perish, Inc. pp. 253–255. ISBN 0-914098-89-6.
  3. ^ Rudin, W. (1976). Principles of Mathematical Analysis (3rd. edition). New York: McGraw-Hill. pp. 120–122. ISBN 007054235X.
  4. ^ Wolfram MathWorld
  5. ^ Spivak 2008, chapter 13.

References

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