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

Free tagged partition 
Given a closed interval  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 
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 
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 
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 
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 
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 
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. If  is iintegrable on  then  is integrable on each subinterval of . If  is integrable on  and  then  is integrable on  and . If  is continuous on  then  is integrable on . If  is monotonous on  then  is integrable on . Let  be a differentiable and strictly monotonous function. Then  is integrable on  if and only if  is integrable on . In such case . If  is integrable on  then  is integrable on  and , for every . 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 [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 [Prop.2.2.3 e Th. 6.1.2]). Theorem 1 (on the absolute integrability of the McShane integral) 
If  is McShane integrable on  then  is also McShane integrable on  and. Theorem 2 (fundamental theorem of Henstock-Kurzweil integral) 
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. Example 3 
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 
The more surprising result of the McShane integral is stated in the following theorem, already announced in the introduction. Theorem 3 
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 [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. Furtherly to the book by Russel Gordon, on this subject we call the attention of the reader also to the works by Robert McLeod [Ch. 8], [Ch. 10] and Douglas Kurtz and Charles Swartz. 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 [Ch.II] or Serge Lang [Ch.X, §4 Appendix] (see also). See also 

 Henstock-Kurzweil integral

References 

Definitions of mathematical integration