In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, is one way to make this concept rigorous and to extend it to more general functions. The Lebesgue integral is more general than the Riemann integral, which it largely replaced in mathematical analysis since the first half of the 20th century. It can accommodate functions with discontinuities arising in many applications that are pathological from the perspective of the Riemann integral. The Lebesgue integral also has generally better analytical properties. For instance, under mild conditions, it is possible to exchange limits with Lebesgue integration, while the conditions for doing this with a Riemann integral are comparatively restrictive. Furthermore, the Lebesgue integral can be generalized in a straightforward way to more general spaces, measure spaces, such as those that arise in probability theory. 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 the Lebesgue measure.
Introduction The integral of a positive real function f between boundaries a and b can be interpreted as the area under the graph of f, between a and b. This notion of area fits some functions, mainly piecewise continuous functions, including elementary functions, for example polynomials. However, the graphs of other functions, for example the Dirichlet function, do not fit well with the notion of area. Graphs like that of the latter raise the question, "For which class of functions does 'area under the curve' make sense?". The answer to this question has great theoretical importance. As part of a general movement toward rigor 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 describes better how and when it is possible to take limits under the integral sign (via the monotone convergence theorem and dominated convergence theorem). While the Riemann integral considers the area under a curve as made out of vertical rectangles, the Lebesgue definition considers horizontal slabs that are not necessarily just rectangles, and so it is more flexible. For this reason, the Lebesgue definition makes it possible to calculate integrals for a broader class of functions. For example, the Dirichlet function, which is 1 where its argument is rational and 0 otherwise, has a Lebesgue integral, but does not have a Riemann integral. Furthermore, the Lebesgue integral of this function is zero, which agrees with the intuition that when picking a real number uniformly at random from the unit interval, the probability of picking a rational number should be zero. Lebesgue summarized his approach to integration in a letter to Paul Montel:
I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral. 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.
Intuitive interpretation
Folland (1999) summarizes the difference between the Riemann and Lebesgue approaches thus: "to compute the Riemann integral of f, one partitions the domain [a, b] into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f". For the Riemann integral, the domain is partitioned into intervals, and bars are constructed to meet the height of the graph. The areas of these bars are added together, and this approximates the integral, in effect by summing areas of the form f(x)dx where f(x) is the height of a rectangle and dx is its width. For the Lebesgue integral, the range is partitioned into intervals, and so the region under the graph is partitioned into horizontal "slabs" (which may not be connected sets). The area of a small horizontal "slab" under the graph of f, of height dy, is equal to the measure of the slab's width times dy:
μ ( { x ∣ f ( x ) > y } ) d y . {\displaystyle \mu \left(\{x\mid f(x)>y\}\right)\,dy.}
The Lebesgue integral may then be defined by adding up the areas of these horizontal slabs. From this perspective, a key difference with the Riemann integral is that the "slabs" are no longer rectangular (cartesian products of two intervals), but instead are cartesian products of a measurable set with an interval.
Simple functions
An equivalent way to introduce the Lebesgue integral is to use so-called simple functions, which generalize the step functions of Riemann integration. Consider, for example, determining the cumulative COVID-19 case count from a graph of smoothed cases each day (right).
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