|
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 -axis. The Lebesgue integral extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined. Mathematicians had long understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the ''area under the curve'' could be defined as the integral, and computed using approximation techniques on the region by polygons. However, as the need to consider more irregular functions arose—e.g., as a result of the limiting processes of mathematical analysis and the mathematical theory of probability—it became clear that more careful approximation techniques were needed to define a suitable integral. Also, we might wish to integrate on spaces more general than the real line. The Lebesgue integral provides the right abstractions needed to do this important job. The Lebesgue integral plays an important role in the branch of mathematics called real analysis, and in many other mathematical sciences fields. It is named after Henri Lebesgue (1875–1941), who introduced the integral . It is also a pivotal part of the axiomatic theory of probability. The term ''Lebesgue integration'' can mean either the general theory of integration of a function with respect to a general measure, as introduced by Lebesgue—or the specific case of integration of a function defined on a sub-domain of the real line with respect to Lebesgue measure. == Introduction == The integral of a function between limits and can be interpreted as the area under the graph of . This is easy to understand for familiar functions such as polynomials, but what does it mean for more exotic functions? In general, for which class of functions does "area under the curve" make sense? The answer to this question has great theoretical and practical importance. As part of a general movement toward rigour in mathematics in the nineteenth century, mathematicians attempted to put integral calculus on a firm foundation. The Riemann integral—proposed by Bernhard Riemann (1826–1866)—is a broadly successful attempt to provide such a foundation. Riemann's definition starts with the construction of a sequence of easily calculated areas that converge to the integral of a given function. This definition is successful in the sense that it gives the expected answer for many already-solved problems, and gives useful results for many other problems. However, Riemann integration does not interact well with taking limits of sequences of functions, making such limiting processes difficult to analyze. This is important, for instance, in the study of Fourier series, Fourier transforms and other topics. The Lebesgue integral is better able to describe how and when it is possible to take limits under the integral sign (via the powerful monotone convergence theorem and dominated convergence theorem). The Lebesgue definition considers a different class of easily calculated areas than the Riemann definition—which is the main reason the Lebesgue integral behaves better. The Lebesgue definition also makes it possible to calculate integrals for a broader class of functions. For example, the Dirichlet function, which is 0 where its argument is irrational and 1 otherwise, has a Lebesgue integral, but does not have a Riemann integral. Lebesgue summarized his approach to integration in a letter to Paul Montel: The insight is that one should be able to rearrange the values of a function freely, while preserving the value of the integral. This process of rearrangement can convert a very pathological function into one that is "nice" from the point of view of integration—and thus let such pathological functions be integrated. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lebesgue integration」の詳細全文を読む スポンサード リンク
|