“Henstock–Kurzweil integral”的意思、由来-开放百科全书

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Henstock–Kurzweil integral

释义

Definition
Properties
McShane integral
See also
References
Footnotes General
External links

In mathematics, the Henstock–Kurzweil integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced {{IPA-fr|dɑ̃ˈʒwa|}}), Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral – is one of a number of definitions of the integral of a function. It is a generalization of the Riemann integral, and in some situations is more general than the Lebesgue integration.

This integral was first defined by Arnaud Denjoy (1912). Denjoy was interested in a definition that would allow one to integrate functions like

This function has a singularity at 0, and is not Lebesgue integrable. However, it seems natural to calculate its integral except over the interval [−ε,δ] and then let ε, δ → 0.

Trying to create a general theory, Denjoy used transfinite induction over the possible types of singularities, which made the definition quite complicated. Other definitions were given by Nikolai Luzin (using variations on the notions of absolute continuity), and by Oskar Perron, who was interested in continuous major and minor functions. It took a while to understand that the Perron and Denjoy integrals are actually identical.

Later, in 1957, the Czech mathematician Jaroslav Kurzweil discovered a new definition of this integral elegantly similar in nature to Riemann's original definition which he named the gauge integral; the theory was developed by Ralph Henstock. Due to these two important contributions, it is now commonly known as the Henstock–Kurzweil integral. The simplicity of Kurzweil's definition made some educators advocate that this integral should replace the Riemann integral in introductory calculus courses.^[1]

Definition

Given a tagged partition P of [a, b], that is,

together with

we define the Riemann sum for a function

to be

where

Given a positive function

which we call a gauge, we say a partition P is -fine if

We now define a number I to be the Henstock–Kurzweil integral of f if for every ε > 0 there exists a gauge such that whenever P is -fine, we have

If such an I exists, we say that f is Henstock–Kurzweil integrable on [a, b].

Cousin's theorem states that for every gauge , such a -fine partition P does exist, so this condition cannot be satisfied vacuously. The Riemann integral can be regarded as the special case where we only allow constant gauges.

Properties

Let {{nowrap|f: [a, b] → R}} be any function.

If {{nowrap|a < c < b}}, then f is Henstock–Kurzweil integrable on [a, b] if and only if it is Henstock–Kurzweil integrable on both [a, c] and [c, b], and then

The Henstock–Kurzweil integral is linear, i.e., if f and g are integrable, and α, β are reals, then αf + βg is integrable and

If f is Riemann or Lebesgue integrable, then it is also Henstock–Kurzweil integrable, and the values of the integrals are the same. The important Hake's theorem states that

whenever either side of the equation exists, and symmetrically for the lower integration bound. This means that if f is "improperly Henstock–Kurzweil integrable", then it is properly Henstock–Kurzweil integrable; in particular, improper Riemann or Lebesgue integrals such as

are also Henstock–Kurzweil integrals. This shows that there is no sense in studying an "improper Henstock–Kurzweil integral" with finite bounds. However, it makes sense to consider improper Henstock–Kurzweil integrals with infinite bounds such as

For many types of functions the Henstock–Kurzweil integral is no more general than Lebesgue integral. For example, if f is bounded with compact support, the following are equivalent:

f is Henstock–Kurzweil integrable,
f is Lebesgue integrable,
f is Lebesgue measurable.

In general, every Henstock–Kurzweil integrable function is measurable, and f is Lebesgue integrable if and only if both f and |f| are Henstock–Kurzweil integrable. This means that the Henstock–Kurzweil integral can be thought of as a "non-absolutely convergent version of Lebesgue integral". It also implies that the Henstock–Kurzweil integral satisfies appropriate versions of the monotone convergence theorem (without requiring the functions to be nonnegative) and dominated convergence theorem (where the condition of dominance is loosened to g(x) ≤ f_n(x) ≤ h(x) for some integrable g, h).

If F is differentiable everywhere (or with countable many exceptions), the derivative F′ is Henstock–Kurzweil integrable, and its indefinite Henstock–Kurzweil integral is F. (Note that F′ need not be Lebesgue integrable.) In other words, we obtain a simpler and more satisfactory version of the second fundamental theorem of calculus: each differentiable function is, up to a constant, the integral of its derivative:

Conversely, the Lebesgue differentiation theorem continues to hold for the Henstock–Kurzweil integral: if f is Henstock–Kurzweil integrable on [a, b], and

then F′(x) = f(x) almost everywhere in [a, b] (in particular, F is almost everywhere differentiable).

The space of all Henstock–Kurzweil-integrable functions is often endowed with the Alexiewicz norm, with respect to which it is barrelled but incomplete.

McShane integral

Lebesgue integral on a line can also be presented in a similar fashion.

First, changing

(here is a -neighbourhood of a) in the notion of -fine partition yields a definition of the Henstock–Kurzweil integral equivalent to the one given above. But after this change we can drop the condition

and get a definition of the McShane integral, which is equivalent to the Lebesgue integral.

References

Footnotes

1. ^{{cite web|title=An Open Letter to Authors of Calculus Books|url=http://www.math.vanderbilt.edu/~schectex/ccc/gauge/letter/|accessdate=27 February 2014}}

General

{{cite book | first = Robert G. |last= Bartle | author-link=Robert G. Bartle| title = A Modern Theory of Integration | series = Graduate Studies in Mathematics |volume = 32 |publisher=American Mathematical Society | year=2001 | isbn=978-0-8218-0845-0}}
[https://www.researchgate.net/publication/308903165_A_modern_Integration_Theory_for_21st_century?ev=prf_pub A Modern Integration Theory in 21st Century]
{{cite book | first1 = Robert G. |last1 = Bartle | author-link=Robert G. Bartle| first2= Donald R. |last2= Sherbert|title = Introduction to Real Analysis |publisher=Wiley |edition=3rd| year=1999 | isbn=978-0-471-32148-4}}
{{cite book | first1=V G |last1=Čelidze |first2= A G |last2= Džvaršeǐšvili|title= The Theory of the Denjoy Integral and Some Applications | series= Series in Real Analysis | volume=3| publisher=World Scientific Publishing Company | year = 1989 |isbn=978-981-02-0021-3}}
{{cite book |first1= A.G. |last = Das|title=The Riemann, Lebesgue, and Generalized Riemann Integrals |publisher=Narosa Publishers |year=2008 |isbn = 978-81-7319-933-2 }}
{{cite book | last=Gordon | first=Russell A. | title=The integrals of Lebesgue, Denjoy, Perron, and Henstock | series=Graduate Studies in Mathematics | volume=4 | publisher=American Mathematical Society | location=Providence, RI | year=1994 | isbn=978-0-8218-3805-1 }}
{{cite book | first=Ralph|last=Henstock| author-link=Ralph Henstock | title=Lectures on the Theory of Integration|series = Series in Real Analysis | volume=1| publisher=World Scientific Publishing Company | year=1988|isbn=978-9971-5-0450-2}}
{{cite book | first = Jaroslav | last= Kurzweil | author-link=Jaroslav Kurzweil | title=Henstock–Kurzweil Integration: Its Relation to Topological Vector Spaces |series = Series in Real Analysis | volume=7| publisher=World Scientific Publishing Company | year = 2000 |isbn=978-981-02-4207-7}}
{{cite book | first = Jaroslav | last= Kurzweil | author-link=Jaroslav Kurzweil | title=Integration Between the Lebesgue Integral and the Henstock–Kurzweil Integral: Its Relation to Locally Convex Vector Spaces|series = Series in Real Analysis | volume=8| publisher=World Scientific Publishing Company | year = 2002 |isbn=978-981-238-046-3}}
{{cite book |first=Solomon |last=Leader |title = The Kurzweil–Henstock Integral & Its Differentials | series= Pure and Applied Mathematics Series | publisher= CRC | year = 2001 | isbn = 978-0-8247-0535-0 }}
{{cite book | first=Peng-Yee |last=Lee | title=Lanzhou Lectures on Henstock Integration|series = Series in Real Analysis | volume=2| publisher=World Scientific Publishing Company | year = 1989 |isbn=978-9971-5-0891-3}}
{{cite book | last1=Lee |first1 = Peng-Yee |last2= Výborný|first2=Rudolf |title = Integral: An Easy Approach after Kurzweil and Henstock |series = Australian Mathematical Society Lecture Series | publisher= Cambridge University Press |year=2000|isbn=978-0-521-77968-5}}
{{cite book | last=McLeod | first=Robert M. | title=The generalized Riemann integral | series=Carus Mathematical Monographs|volume=20 | publisher=Mathematical Association of America | location=Washington, D.C. | year=1980 | isbn=978-0-88385-021-3 }}
{{cite book | last=Swartz | first=Charles W. |title=Introduction to Gauge Integrals | publisher= World Scientific Publishing Company |year=2001 | isbn=978-981-02-4239-8 }}
{{cite book | first1= Charles W. |last1= Swartz| first2 =Douglas S. | last2=Kurtz |title= Theories of Integration: The Integrals of Riemann, Lebesgue, Henstock–Kurzweil, and McShane|series= Series in Real Analysis |volume=9| publisher= World Scientific Publishing Company | year = 2004 |isbn = 978-981-256-611-9 }}

External links

The following are additional resources on the web for learning more:

{{springer|title=Kurzweil-Henstock integral|id=p/k110200}}
An Introduction to The Gauge Integral
An Open Suggestion: To replace the Riemann integral with the gauge integral in calculus textbooks signed by Bartle, Henstock, Kurzweil, Schechter, Schwabik, and Výborný

1 : Definitions of mathematical integration

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