- Lebesgue integration
mathematics, the " integral" of a non-negative function can be regarded in the simplest case as the areabetween the graph of that function and the "x"-axis. Lebesgue integration is a mathematical construction that extends the integralto a larger class of functions; it also extends the domains on which these functions can be defined. It had long been 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 techniques of approximation of the region by polygons. However, as the need to consider more irregular functions arose (for example, as a result of the limiting processes of mathematical analysisand the mathematical theory of probability) it became clear that more careful approximation techniques would be needed in order to define a suitable integral.
The Lebesgue integral plays an important role in the branch of mathematics called
real analysisand in many other fields in the mathematical sciences.
The Lebesgue integral is named for
Henri Lebesgue( 1875- 1941) who introduced the integral in harv|Lebesgue|1904. His last name is pronounced|ləˈbɛg, approximately "luh beg".
The term "Lebesgue integration" may refer either to the general theory of integration of a function with respect to a general measure, as introduced by Lebesgue, or to the specific case of integration of a function defined on a sub-domain of the
real linewith respect to Lebesgue measure.
The integral of a function "f" between limits "a" and "b" can be interpreted as the area under the graph of "f". This is easy to understand for familiar functions such as
polynomials, but what does it mean for more exotic functions? In general, what is the class of functions for which "area under the curve" makes sense? The answer to this question has great theoretical and practical importance.
As part of a general movement toward
rigourin mathematics in the nineteenth century, attempts were made to put the 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 for the integral. Riemann's definition starts with the construction of a sequence of easily-calculated areas which 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 of prime importance, 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. The Lebesgue definition considers a different class of easily-calculated areas than the Riemann definition, which is the main reason the Lebesgue integral is better behaved.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 it does not have a Riemann integral.
Construction of the Lebesgue integral
The discussion that follows parallels the most common expository approach to the Lebesgue integral. In this approach, the theory of integration has two distinct parts:
# A theory of measurable sets and measures on these sets.
# A theory of measurable functions and integrals on these functions.
Measure theoryinitially was created to provide a detailed analysis of the notion of length of subsets of the real line and more generally area and volume of subsets of Euclidean spaces. In particular, it provided a systematic answer to the question of which subsets of R have a length. As was shown by later developments in set theory(see non-measurable set), it is actually impossible to assign a length to all subsets of R in a way which preserves some natural additivity and translation invariance properties. This suggests that picking out a suitable class of "measurable" subsets is an essential prerequisite.
Of course, the Riemann integral uses the notion of length implicitly. Indeed, the element of calculation for the Riemann integral is the rectangle ["a", "b"] × ["c", "d"] , whose area is calculated to be ("b"−"a")("d"−"c"). The quantity "b"−"a" is the length of the base of the rectangle and "d"−"c" is the height of the rectangle. Riemann could only use planar rectangles to approximate the area under the curve because there was no adequate theory for measuring more general sets.
In the development of the theory in most modern textbooks (after 1950), the approach to measure and integration is "axiomatic". This means that a measure is any function μ defined on certain subsets "X" of a set "E" which satisfies a certain list of properties. These properties can be shown to hold in many different cases.
The theory of measurable sets and measure (including definition and construction of such measures) is discussed in other articles. See measure.
For example, "E" can be Euclidean "n"-space R"n" or some Lebesgue measurable subset of it, "X" will be the σ-algebra of all Lebesgue measurable subsets of "E", and μ will be the
Lebesgue measure. In the mathematical theory of probability μ will be a probabilitymeasure on a probability space "E".
In Lebesgue's theory, integrals are limited to a class of functions called
measurable functions. A function "f" is measurable if the pre-image of every closed interval is in "X":
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