Henstock–Kurzweil integral

The gauge integral ( also: calibration integral, Henstock integral, Henstock - Kurzweil integral) is an integral type whose current formulation was not discovered until the mid-20th century by the mathematician Jaroslav Kurzweil. Ralph Henstock devoted himself to the development of the theory of this integral type. A central estimate, called the Henstock lemma is named after him. " Precursor " is the (equivalent) Dejoy -Perron integral, which however, is based on a very technical and non-intuitive definition.

The peculiarity of the gauge integral is that every derivation function automatically ( ie without additional assumptions ) is integrable with. Besides occur in the theory of gauge integral conditionally integrable functions. This refers to functions which, although integrable, but not the amount. Both the Riemann and in the Lebesgue definition follows from the integration of a function is always the integrability of its amount.

The gauge integral contains both the Riemann and the Lebesgue integral as special cases, ie, each Riemann - Lebesgue integrable function and is gauge integrable; However, since there are functions that are not yet gauge - integrable but Lebesgue integrable and Riemann, is the gauge integral a real extension of the Lebesgue integral dar.

The name Eich integral ( Gauge is the English term for calibration) owes the integral of its definition: Similar to the Riemann integral come when calibration integral decompositions and Riemann sums are used, the fineness of a decomposition is, however, a special intervallwertigen feature called calibration function evaluated.

• 2.1 preliminary work
• 2.2 Example
• 2.3 Definition of the gauge integral

• 4.1 extensions in one dimension
• 4.2 The multi-dimensional gauge integral

Introduction

The main theorem

The fundamental theorem of differential and integral calculus ( in the current count to be 1 part ) is a central theorem in the theory of Riemann and Lebesgue integral. It reads:

• Sentence: If a derivation function over the interval of Riemann (or Lebesgue ) integrable, then:.

The main theorem gives in practice one of the most important methods to determine the value of an integral concrete and precise. If you want to about the function by integrating over, so we group f as the derivative of a function, called the primitive function on. Apparently is given by a primitive function of, so that follows:

In both the Riemann and the Lebesgue integral, however, the integration of must be cited as a requirement - not every derivation function is necessarily integrable. Rather indicates that there are derivative functions not Riemann and not integrable Lebesgue. An example is the function with

( see Figure 1). Its derivative is by

Given. Since it is not limited, is not Riemann integrable -. It shows that is not Lebesgue integrable.

A ( descriptive ) analysis of the reasons for which is not Riemann - integrable, leading to a decisive improvement in the Riemann definition. These considered at the initial stage, where the formula comes at all.

The straddle - lemma and the problems of the Riemann integral

By the mean value theorem of differential calculus there is a differentiable function on an interval with a

If we choose a decomposition intermediate points by the mean value theorem, we obtain the result for Riemann sums:

The last sum presented here a telescopic sum dar. For other intermediaries applies in the above statement iA no equality, but for the proof of it is not necessary that all Riemann sums are exactly alike. It is sufficient that the Riemann sums for any number of intermediate points approaching arbitrarily, as long as you only sufficiently fine selects the considered decompositions. This would for instance be the case when a function of each interval, the approximation for all

Met, the resultant by the approximation error is as small as desired, provided that the interval just sufficient small ( Fig. 2).

But there are features that just do not show this behavior. One such feature is the function from the previous section. Take, for instance, the interval for any (even arbitrarily small ) oscillates near 0 " wildly ", therefore can be applied to each interval of this form (no matter how small it was, too) find a job, so that an arbitrarily large positive or negative number. The average slope over the interval, however, tends to 0 when tends to 0. Finally, and the average slope of g over the interval just the difference quotient of at the point 0:

So can deviate arbitrarily strongly from the average slope on the interval. Since each partition Z is an interval of the form "contains", there is a decomposition for each sub-interval and certain intermediate points, for which the approximation is violated. This can - as in the case of the function g - cause is not Riemann integrable, because according to the Riemann definition yes all intermediate points need to be investigated to a partition Z. An integral definition would be desirable in dealing with specific intervals only certain intermediaries may also be considered. For the purpose of integration of the function it would be, for example, helpful to allow only the intermediate point 0 for the subinterval, because after approximation ( 1) would be fulfilled.

An integration theory based on Riemann sums and in each derivation function is integrable, should according to the previous considerations into account only those pairs of decompositions and intermediate points, for the

Applies. The following theorem opened a way to identify such pairs:

• Set ( Straddle Lemma ): Let differentiable in. Then for each a with and for all with.

If you divide the inequality of the straddle lemma by his core message is apparently: for each point a closed interval for which

Is correct. The number indicates the error in the approximation. Since arbitrarily, ie in particular arbitrarily small, must be an interval can even be found always on the above approximation is as good as any. The only requirement is that the interval boundaries x and y are sufficiently close formulated differently in, or: The prerequisite is that the interval lies in a sufficiently small neighborhood of t:

If we choose only those pairs from cutting out along with intermediate points, for which the condition

( being chosen according to the straddle - lemma) is true, then the approximation is always satisfied, and all of its Riemann sums are in close as desired.

It raises the question of how to choose these " appropriate " combinations of all possible combinations of intermediate points and decompositions. The Riemann fineness term, ie the viewing of the largest interval length, is not fit to do so. Obviously go to the selected intermediate points and thus the position of the sub-intervals not included in the evaluation of the fineness of the decomposition. The relevant number from the straddle - Lemma but iA depend upon the location t! One example is expect that the smaller, the more f oscillates in the vicinity of this point. Therefore, it may well happen that for a decomposition and intermediate points, the condition is satisfied, for an equally fine dissection but not (see Figure 3), even not if the same intermediate point is considered. The aim will therefore be to provide an improved fineness term which takes into account the position of the sub-intervals.

Basic ideas

In summary, are the " guidelines" for the definition of the gauge integral:

• As part of a new integral type each derivative function automatically (ie without additional assumptions ) should be integrated with.
• But the relationship between intermediaries and decompositions must be regulated so that it is possible to combine intermediate points with such decompositions, the " good match ". For this purpose, a refinement term should be created, the the position of the sub-intervals, considered and
• Which makes it possible to admit to certain subintervals only certain intermediaries.

The formal definition

Preparatory work

As for the new Integral only to each other " appropriate" decompositions and intermediate points to be considered, it is natural to merge the two concepts in a first term.

• Definition ( marked decomposition ). Be a decomposition of an interval and corresponding to Z intermediaries, ie it applies to. The quantity is called a marked separation (English: tagged partition ) of the interval.

Thus, a marked decomposition comprises two- tuples of the form, where I is a closed interval, and t is a number having. Riemann sums with respect to a function f and a marked decomposition D is defined exactly by how Riemann subtotals:

The following definition provides the basis for an improved fineness term:

• Definition ( calibration function ): A intervallwertige function on the interval is called calibration function if and an open interval.

So a calibration function assigns each point to an open interval that contains t. On the Concept of the calibration function can now be a very flexible Feinheitsmaß define that takes into account not only the position of the subintervals of a decomposition Z, but on the can also regulate the relationship between decomposition and intermediaries: A marked decomposition D is then fine hot when a calibration function, and each sub-interval located within that open interval supplies belonging to the Sub- interval intermediate point:

• Definition: Let a calibration function on the interval [a, b] and a marked decomposition of this interval. D is called - fine if for all.

Example

Limitation to fine decompositions it is - by a clever choice of the calibration function - possible to select only matching pairs of decompositions and interpolation points. Be about Z and a partition of this interval. Is (as in the example of the function g ') which are admitted to 0 only possible intermediary part interval is defined as follows:

And it being arbitrary. Then the only given by open interval containing the 0. For each marked decomposition D of [0,1] but must apply:. Since a marked separation can only be fine if. The subinterval therefore occurs in any fine marked decomposition on exclusively with the intermediate point 0. Furthermore, the " smallness " of a sub-interval of a selected separation D in response to the intermediate position and thus "tuned" by the position of the sub-interval due to the T- dependency of the function.

Definition of the gauge integral

The gauge integral is now - much like the Riemann integral - defined as a fixed number, the Riemann sums with respect to labeled partitions of an interval approach to any size, provided that such decompositions fine regarding appropriate calibration functions can be selected:

• Definition ( gauge integral): A function is called gauge - integrable ( eichintegrabel, Henstock ( Kurzweil ) integrable ) on [a, b ] if there is a fixed number to each a calibration function on [a, b] such that applies to any fine marked decomposition D. called gauge integral ( calibration integral, Henstock ( Kurzweil ) integral) of f over [a, b], in characters.

The definition is strongly reminiscent of the (original ) definition of the Riemann integral. The important difference is that the coarse Riemann Feinheitsmaß was replaced ( observation interval of the longest part of the partition Z ) by the new and improved degree. Henstock therefore speaks in his book Theories of Integration, by a " integral of Riemann - type".

Properties of the gauge integral

As with any other integral type shall be considered:

• The value of the gauge integral is uniquely determined.

Furthermore, the integral function is linear:

• If two function on [a, b] Gauge - integrable and, then also gauge integrable is on [a, b] and we have:.

The Riemann integral fits naturally into the framework of the gauge integral:

• Each Riemann - integrable function is gauge integrable and the two integrals coincide.

Be to the Riemann integral of f over [a, b] and chosen such that for every partition Z and any intermediate points. If you select the calibration function to

Then for every fine marked decomposition by definition: ie. If we define the partition Z, it is and therefore:

Also known by the Riemann and Lebesgue integral Intervalladditivität applies:

• Be and two non-overlapping closed intervals ( ie, the two intervals have at most one boundary point in common ) and f over gauge integrable. Then f is also gauge integrable and:.

Conversely, one finds:

• Is on the non- overlapping intervals gauge integrable. If, then f is also integrable and we have:

The gauge integral is monotonic:

• Is gauge integrable over and (that ), then:

It is particularly interesting that every derivation function is gauge integrable:

• ( Main clause, part 1) Be differentiable. Then, with over gauge - integrable.

The result is obtained after a few deft transformations by choosing to the (symmetric ) calibration function, which is determined by factors straddle lemma. Then one evaluates the expression for an arbitrary fine - marked decomposition. The second part of the main clause is for the gauge integral:

• ( Main clause, part 2) Let gauge integrable over. Then the function with almost everywhere in [a, b] is differentiable.

So it is for the indefinite integral of a gauge - integrable function, the statement " F is not differentiable or it is " at most on a Lebesgue -null set correctly. It is important that only the integrability of f must be provided. If f is even continuous, then F is everywhere in [a, b], differentiable.

For the gauge integral, the two central, known by the Lebesgue integral Konvergenztheoreme apply. These describe the circumstances under which the limit function of a sequence of functions integrable functions gauge again gauge is integrable and integration and thresholding may be interchanged:

Is obtained:

• Theorem on monotone convergence: Let I be an interval, a sequence of functions which are integrable and I gauge. Converges monotonically to, and that is true for all, it is precisely then gauge integrable over when. In this case, the following applies:

Converges a sequence of functions pointwise to a limit function and is the result for each and every monotone increasing function on I gauge integrable, the limit function is then and only then I gauge integrable if the sequence is bounded. In this case, the integration and thresholding may be reversed, both operations are thus performed in reverse order.

Also applies the

• Theorem on dominated convergence: Let I be an interval, a sequence of functions which are integrable and I gauge. Converges pointwise to and there are gauge integrable functions with almost everywhere in I and all, then f is integrable on I gauge and we have:

So is there an over I gauge integrable minorant and over I gauge integrable majorant for, so the limit function f of the function sequence gauge integrable on I. Also in this case, limit education and integration may be interchanged.

Extensions

The following is always to understand Lebesgue measurability, the term measurability (and correspondingly related terms ). The measure is thus looked to the Lebesgue measure.

Extensions in one dimension

The gauge integral can be extended to infinite intervals. At first, this seems surprising. Looking at the interval as an example, it is first faced with the problem that the interval is not closed. This problem can be easily addressed by not, but sets the extended real numbers based. Analogues, in the integration over each open interval before: you then always regarded the conclusion of the interval, ie, the closed interval [a, b], taking also and / or approved. Thus, but the problems are far from resolved: Since the gauge integral works with finite decompositions, in the case of an infinite integration region at least a portion of each labeled interval decomposition of I is infinite ( or either or both ), and thus the sum

At best, infinity, and at worst not even defined when two infinitely long intervals occur and f values ​​with different signs takes to the respective intermediate points ( then enters the undefined expression on ). One could now similar to the Riemann integral define improper integrals, but it turns out that this is not necessary by using a trick: this is examined in the case of an infinite definition interval I is not the integral of f, but about, given by:

In particular:. Within the Riemann sum S (f, D) is then the Convention apply. Thus, each Riemann sum S (f, D) is also defined if D contains infinitely long intervals, so far this only occur with the intermediate locations together. However, this can force through the following definition:

• Definition: The interval is open interval containing. The analogy with the open interval containing.

It is now possible to define calibration functions so that infinitely long subintervals only occur together with an intermediate places, eg for the interval:

Where a, b are arbitrary real numbers, and R and L may be any positive real function. Da and the only intervals from the range of values ​​of are that are infinitely long, the sub-interval of a fine marked decomposition D can occur because the condition only with the intermediate point along. The same applies for the sub- interval can only occur together with said intermediate point. The example of the decomposition

And a function becomes clear why thus the problem of the infinite / undefined Riemann sums is solved:

The two potentially infinite summands omitted and the Riemann sum is finite. With these new definitions, the gauge integral can be easily extended to infinite and / or open subintervals:

• Definition: Let any interval, and his degree in (ie, there are also and authorized ). f is called gauge integrable ( Henstock ( Kurzweil ) - integrable, eichintergrabel ) above, if there is a fixed number to each a calibration function, so that for each fine marked decomposition decomposition D of. It is called A, the gauge integral of f over I, in characters.

Is any measurable subset of an interval I, it is called gauge - integrable on E if the function I gauge is integrable. One then defines the gauge integral of f over E by:

If a measurable amount and a measurable function, then f is called integrable gauge on when the extension of f to, so the function with

Is gauge integrable and sets:

It shows:

• If we define improper gauge integral similar to the improper integrals in the Riemann theory, so f is improperly gauge integrated over an infinitely long interval of definition when it is actually gauge integrable in the above sense, also tune the values ​​of the integrals coincide.
• Transfer all properties mentioned in the previous section,. Mutatis mutandis to the extended definition to infinite intervals gauge integral The first part of the main clause applies then on every finite subinterval of an infinitely long integration region I, in the second part an arbitrary fixed point must be selected. The content of the theorem is then valid for the function, which is possible.

Due to the fall Intervalladditivität all extended definitions with the original definition of the gauge integral over intervals together, if E is an interval ( each interval is measurable). With these definitions, the connection succeeds to the Lebesgue integral. It shows:

• Be a measurable amount. Is Lebesgue integrable on E, then also gauge integrable is E and the two integrals coincide. In particular: f is Lebesgue integrable on E if f is absolutely integrable gauge on E, ie, both the function f and the amount is above E gauge integrable are.

So that is also Lebesgue integral " as a special case in gauge Integral".

The multidimensional integral gauge

Correspondingly, the gauge integral is continued to arbitrary dimensions. As in one dimension the integral first defined on this interval. The extension to infinite intervals to be included therein.

• Definition ( n-dimensional interval ) A set is ( n-dimensional ) interval when there intervals.

An interval in n- dimensions is thus defined as the Cartesian product n one-dimensional intervals and consequently has the shape of an n-dimensional cube. The following applies:

• Definition ( open, closed): An interval is called [ closed ] openly, if all open [ closed ] are.

Note that a range of the form, with or is referred to as closed.

Accordingly, we extended the terms of the marked decomposition and the calibration function on dimensions:

• Definition ( marked decomposition ): Let be a closed interval. A set is marked decomposition of I, if all intervals with and.

A marked decomposition of a closed interval is therefore from a set of two- tuples whose first entry is a point whose second entry, however, is an interval. The belonging to the interval point has to lie in the union of all in turn yield the interval to be separated ( see Figure 8).

• Definition ( calibration function ): Let an interval. A intervallwertige function is called calibration function on I if an open interval and for all.

As in the one-dimensional to an interval shall also be open when one has the shape or with an arbitrary real number. Just as in one dimension is now defined using these terms, the fineness of a labeled decomposition:

• Definition: Let a calibration function on the closed interval I. A marked decomposition is called fine if for.

The volume of an interval is given by:

Wherein the length of ( one-dimensional ) interval represents. Again, to apply the Convention, ie, has one of the length 0, so even if one or more infinitely long intervals under the have.

Each function will continue to:

Especially disappears at each point has at least one endless component. So is about. Riemann subtotals with respect to a function and a marked decomposition are defined by:

Again, the convention was valid. The gauge integral in dimensions can then be as follows:

• Definition ( n-dimensional gauge integral): Let I be an interval of whose financial statements in. f is called gauge integrable ( Henstock ( Kurzweil ) intergrabel, eichintegrabel ) on I if there is a fixed number to each a calibration function, such that for every subtle marked decomposition D: . One writes: .

All extensions to arbitrary measurable subsets of done the same way as in the one-dimensional gauge integral. The above-mentioned Properties of one-dimensional gauge integral carry over mutatis mutandis to the multi-dimensional gauge integral. In addition, versions of the sets of Fubini and Tonelli let up for the n -dimensional gauge integral.

Literature and links

• Integral calculus