In the branch of mathematics known as integration theory, the McShane integral, created by Edward J. McShane,[1] is a modification of the Henstock-Kurzweil integral.[2] The McShane integral is equivalent to the Lebesgue integral.[3]

Definition

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Free tagged partition

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Given a closed interval [a, b] of the real line, a free tagged partition   of   is a set

 

where

 

and each tag  .

The fact that the tags are allowed to be outside the subintervals is why the partition is called free. It's also the only difference between the definitions of the Henstock-Kurzweil integral and the McShane integral.

For a function   and a free tagged partition  , define  

Gauge

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A positive function   is called a gauge in this context.

We say that a free tagged partition   is  -fine if for all  

 

Intuitively, the gauge controls the widths of the subintervals. Like with the Henstock-Kurzweil integral, this provides flexibility (especially near problematic points) not given by the Riemann integral.

McShane integral

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The value   is the McShane integral of   if for every   we can find a gauge   such that for all  -fine free tagged partitions   of  ,

 

Examples

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It's clear that if a function   is integrable according to the McShane definition, then   is also Henstock-Kurzweil integrable. Both integrals coincide in the regard of its uniqueness.

In order to illustrate the above definition we analyse the McShane integrability of the functions described in the following examples, which are already known as Henstock-Kurzweil integrable (see the paragraph 3 of the site of this Wikipedia "Henstock-Kurzweil integral").

Example 1

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Let   be such that   and   if  

As is well known, this function is Riemann integrable and the correspondent integral is equal to   We will show that this   is also McShane integrable and that its integral assumes the same value.

For that purpose, for a given  , let's choose the gauge   such that   and   if  

Any free tagged partition   of   can be decomposed into sequences like

 , for  ,

 , for  , and

 , where  , such that    

This way, we have the Riemann sum

 

and by consequence

 

Therefore if   is a free tagged  -fine partition we have

 , for every  , and

 , for every  .

Since each one of those intervals do not overlap the interior of all the remaining, we obtain

 

Thus   is McShane integrable and

 

The next example proves the existence of a distinction between Riemann and McShane integrals.

Example 2

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Let   the well known Dirichlet's function given by

 

which one knows to be not Riemann integrable. We will show that   is integrable in the MacShane sense and that its integral is zero.

Denoting by   the set of all rational numbers of the interval  , for any   let's formulate the following gauge

 

For any  -fine free tagged partition   consider its Riemann sum

 .

Taking into account that   whenever   is irrational, we can exclude in the sequence of ordered pairs which constitute  , the pairs   where   is irrational. The remainder are subsequences of the type   such that  ,   Since each one of those intervals do not overlap the interior of the remaining, each one of these sequences gives rise in the Riemann sum to subsums of the type

 .

Thus  , which proves that the Dirichlet's function is McShane integrable and that

 

Relationship with Derivatives

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For real functions defined on an interval  , both Henstock-Kurzweil and McShane integrals satisfy the elementary properties enumerated below, where by   we denote indistinctly the value of anyone of those inetegrals.

  1. If   is integrable on   then   is integrable on each subinterval of  .
  2. If   is integrable on   and   then   is integrable on   and  .
  3. If   is continuous on   then   is integrable on  .
  4. If   is monotonous on   then   is integrable on  .
  5. Let   be a differentiable and strictly monotonous function. Then   is integrable on   if and only if   is integrable on  . In such case  .
  6. If   is integrable on   then   is integrable on   and  , for every  .
  7. Let   and   be integrable on  . Then:
    •   is integrable on   and  .
    •   em   .

With respect to the integrals mentioned above, the proofs of these properties are identical excepting slight variations inherent to the differences of the correspondent definitions (see Washek Pfeffer[4] [Sec. 6.1]).

This way a certain parallelism between the two integrals is observed. However an imperceptible rupture occurs when other properties are analysed, such as the absolute integrability and the integrability of the derivatives of integrable differentiable functions.

On this matter the following theorems hold (see[4] [Prop.2.2.3 e Th. 6.1.2]).

Theorem 1 (on the absolute integrability of the McShane integral)

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If   is McShane integrable on   then   is also McShane integrable on   and .

Theorem 2 (fundamental theorem of Henstock-Kurzweil integral)

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If   is differentiable on  , then   is Henstock-Kurzweil integrable on   and .

In order to illustrate these theorems we analyse the following example based upon Example 2.4.12.[4]

Example 3

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Let's consider the function:

 

  is obviously differentiable at any   and differentiable, as well, at  , since  .

Moreover

 

As the function

 

is continuous and, by the Theorem 2, the function   is Henstock-Kurzweil integrable on   then by the properties 6 and 7, the same holds to the function

 

But the function

 

is not integrable on   for none of the mentioned integrals.

In fact, otherwise, denoting by   anyone of such integrals, we should have necessarily   for any positive integer  . Then through the change of variable  , we should obtain taking into account the property 5:

 

 .

As   is an arbitrary positive integer and  , we obtain a contradiction.

From this example we are able to conclude the following relevant consequences:

  • I) Theorem 1 is no longer true for Henstock-Kurzweil integral since   is Henstock-Kurzweil integrable and   is not.
  • II) Theorem 2 does not hold for McShane integral. Otherwise   should be McShane integrable as well as   and by Theorem 1, as  , which is absurd.
  • III)   is, this way, an example of a Henstock-Kurzweil integrable function which is not McShane integrable. That is, the class of McShane integrable functions is a strict subclass of the Henstock-Kurzweil integrable functions.

Relationship with Lebesgue Integral

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The more surprising result of the McShane integral is stated in the following theorem, already announced in the introduction.

Theorem 3

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Let  . Then

  is McShane integrable     is Lebesgue integrable.

The correspondent integrals coincide.

This fact enables to conclude that with the McShane integral one formulates a kind of unification of the integration theory around Riemann sums, which, after all, constitute the origin of that theory.

So far is not known an immediate proof of such theorem.

In Washek Pfeffer[4] [Ch. 4] it is stated through the development of the theory of McShane integral, including measure theory, in relationship with already known properties of Lebesgue integral. In Charles Swartz[5] that same equivalence is proved in Appendix 4.

Furtherly to the book by Russel Gordon[3] [Ch. 10], on this subject we call the attention of the reader also to the works by Robert McLeod[6] [Ch. 8] and Douglas Kurtz together with Charles W. Swartz.[2]

Another perspective of the McShane integral is that it can be looked as new formulation of the Lebesgue integral without using Measure Theory, as alternative to the courses of Frigyes Riesz and Bela Sz. Nagy[7] [Ch.II] or Serge Lang[8] [Ch.X, §4 Appendix] (see also[9]).

See also

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References

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  1. ^ McShane, E. J. (1973). "A Unified Theory of Integration". The American Mathematical Monthly. 80 (4): 349–359. doi:10.2307/2319078. ISSN 0002-9890.
  2. ^ a b Kurtz, Douglas S. and Swartz, Charles W. (2012). Theories of integration: the integrals of Riemann, Lebesgue, Henstock-Kurzweil, and McShane (2nd ed.). Singapore: World Scientific. p. 247. ISBN 978-981-4368-99-5. OCLC 769192118.{{cite book}}: CS1 maint: multiple names: authors list (link)
  3. ^ a b Gordon, Russell A. (1994). The integrals of Lebesgue, Denjoy, Perron, and Henstock. Providence, R.I.: American Mathematical Society. pp. 157–163. ISBN 0-8218-3805-9. OCLC 30474120.
  4. ^ a b c d Pfeffer, Washek F. (1993). The Riemann Approach to Integration. New York: Cambridge University Press. ISBN 0-521-44035-1.
  5. ^ Swartz, Charles (2001). Introduction to Gauge Integrals. World Scientific. ISBN 9810242395.
  6. ^ McLeod, Robert M. (1980). The Generalized Riemann Integral. U. S. A.: The Mathematical Association of America. ISBN 0-88385-000-1.
  7. ^ Riesz, Frigys e Sz.-Nagy, Béla (1990). Functional Analysis. New York: Dover. ISBN 0-486-66289-6.
  8. ^ Lang, Serge (1983). Undergraduate Analysis. New York: Springer-Verlag. ISBN 978-1-4419-2853-5.
  9. ^ Lang, Serge (2012). Real and Functional Analysis (3rd. Edition). Berlin, Heidelberg: Springer-Verlag. ISBN 978-1-4612-6938-0.