Lebesgue integral

Lebesgue integral
an integral obtained by application of the theory of measure and more general than the Riemann integral.
[named after H. L. LEBESGUE]

* * *

      way of extending the concept of area inside a curve to include functions that do not have graphs representable pictorially. The graph of a function is defined as the set of all pairs of x- and y-values of the function. A graph can be represented pictorially if the function is piecewise continuous, which means that the interval over which it is defined can be divided into subintervals on which the function has no sudden jumps. Because the Riemann integral is based on the Riemann sums, which involve subintervals, a function not definable in this way will not be Riemann integrable.

      For example, the function that equals 1 when x is rational and equals 0 when x is irrational has no interval in which it does not jump back and forth. Consequently, the Riemann sum

f (c1x1 + f (c2x2 +⋯+ f (cnxn
has no limit but can have different values depending upon where the points c are chosen from the subintervals Δx.

      Lebesgue sums are used to define the Lebesgue integral of a bounded function by partitioning the y-values instead of the x-values as is done with Riemann sums. Associated with the partition

{yi} (= y0, y1, y2,…, yn)
are the sets Ei composed of all x-values for which the corresponding y-values of the function lie between the two successive y-values yi − 1 and yi. A number is associated with these sets Ei, written as m(Ei) and called the measure of the set, which is simply its length when the set is composed of intervals. The following sums are then formed:
S = m(E0)y1 + m(E1)y2 +⋯+ m(En − 1)yn
and
s = m(E0)y0 + m(E1)y1 +⋯+ m(En − 1)yn − 1.
As the subintervals in the y-partition approach 0, these two sums approach a common value that is defined as the Lebesgue integral of the function.

      The Lebesgue integral is the concept of the measure of the sets Ei in the cases in which these sets are not composed of intervals, as in the rational/irrational function above, which allows the Lebesgue integral to be more general than the Riemann integral.

* * *


Universalium. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Lebesgue-Integral —   [lə bɛg ; nach H. L. Lebesgue], für beschränkte Funktionen auf beschränkten Intervallen eine weitgehende Verallgemeinerung des riemannschen Integralbegriffs (Integralrechnung). Es gibt Funktionen, die Lebesgue integrierbar, aber nicht Riemann… …   Universal-Lexikon

  • Lebesgue-Integral — Illustration der Grenzwertbildung beim Riemann Integral (blau) und beim Lebesgue Integral (rot) Das Lebesgue Integral (nach Henri Léon Lebesgue) ist der Integralbegriff der modernen Mathematik, der die Berechnung von Integralen in beliebigen… …   Deutsch Wikipedia

  • Lebesgue integral — noun /ləˈbɛːɡ ˈɪntəɡrəl,ləˈbɛːɡ ˈɪntəɡrl̩/ An integral which has more general application than that of the Riemann integral, because it allows the region of integration to be partitioned into not just intervals but any s for which the function to …   Wiktionary

  • Lebesgue integral — Math. an integral obtained by application of the theory of measure and more general than the Riemann integral. [named after H. L. LEBESGUE] …   Useful english dictionary

  • Lebesgue — ist der Name folgender Personen: Henri Léon Lebesgue (1875–1941), französischer Mathematiker Octave Lebesgue (1857–1933), französischer Journalist Siehe auch: Lebesgue Maß Lebesgue Integral Lebesgue’sche Überdeckungsdimension Lebesguezahl Satz… …   Deutsch Wikipedia

  • Integral — ist: in der Analysis ein Grenzwert, zum Beispiel zur Berechnung von Flächen und Volumina, siehe Integralrechnung in der Maßtheorie der erweiterte Integralbegriff, siehe Lebesgue Integral in Mathematik und Physik allgemein die Lösung einer… …   Deutsch Wikipedia

  • Lebesgue-integrierbar — Das Lebesgue Integral (nach Henri Léon Lebesgue) ist der Integralbegriff der modernen Mathematik, der die Berechnung von Integralen in beliebigen Maßräumen ermöglicht. Im Fall der reellen Zahlen mit dem Lebesgue Maß stellt das Lebesgue Integral… …   Deutsch Wikipedia

  • Integral — This article is about the concept of integrals in calculus. For the set of numbers, see integer. For other uses, see Integral (disambiguation). A definite integral of a function can be represented as the signed area of the region bounded by its… …   Wikipedia

  • Lebesgue integration — In mathematics, the integral of a non negative function can be regarded in the simplest case as the area between the graph of that function and the x axis. Lebesgue integration is a mathematical construction that extends the integral to a larger… …   Wikipedia

  • Lebesgue-Stieltjes integration — In measure theoretic analysis and related branches of mathematics, Lebesgue Stieltjes integration generalizes Riemann Stieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure theoretic framework.… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”