LEBESGUE INTEGRATION Meaning and
Definition
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Lebesgue integration is a mathematical concept used to define a more general and comprehensive approach to measuring the size of subsets of the real numbers or any abstract space. It was introduced by the French mathematician Henri Lebesgue in the early 20th century as an extension of the traditional Riemann integral.
In the context of real analysis, Lebesgue integration over a set A involves dividing the set A into smaller subsets or intervals and assigning a numerical value to each interval. Unlike the Riemann integral, the Lebesgue integral does not require the function being integrated to be continuous or bounded. It can handle a much wider class of functions, including those that are not defined on an interval.
Lebesgue integration takes into consideration the "measure" of a set, a concept that generalizes the length, area, or volume of a subset. The measure theory provides a systematic way to assign measures to sets based on their properties.
The Lebesgue integral is defined by dividing the range of a function into a sequence of increasingly smaller intervals and measuring the function's value on each interval. The integral is then obtained by summing up these values, weighted by the measures of the intervals.
The Lebesgue integration theory has proved to be a powerful tool in numerous areas of mathematics, including probability theory, functional analysis, and harmonic analysis. It provides a rigorous foundation for integral calculus and enables the analysis of more complex functions and sets.
Etymology of LEBESGUE INTEGRATION
The word "Lebesgue" in "Lebesgue integration" is derived from the name of the mathematician Henri Lebesgue (1875-1941), who developed the theory of Lebesgue integration in the early 20th century. Lebesgue integration extends the concept of integration developed by earlier mathematicians such as Riemann. It provides a more general and powerful framework for integrating a wider range of functions, including those that are not necessarily continuous. Lebesgue's work in this area was a significant advancement in the field of mathematical analysis and has had a profound impact on various branches of mathematics, including probability theory and measure theory.
