Lebesgue integration

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.

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:

, Source: (Siegmund-Schultze 2008)

Condensed: the full section is in Wikipedia.

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: \[\mu \left (\{x\mid f(x)>y\} \right ) \,dy.\] 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.

One can think of the Lebesgue integral either in terms of slabs or simple functions. Intuitively, the area under a simple function can be partitioned into slabs based on the (finite) collection of values in the range of a simple function (a real interval). Conversely, the (finite) collection of slabs in the undergraph of the function can be rearranged after a finite repartitioning to be the undergraph of a simple function.

The slabs viewpoint makes it easy to define the Lebesgue integral, in terms of basic calculus. Suppose that f is a (Lebesgue measurable) function, taking non-negative values (possibly including +∞). Define the distribution function of f as the "width of a slab", i.e., \[F(y) = \mu\{x | f(x) > y\}.\] Then F(y) is monotone decreasing and non-negative, and therefore has an (improper) Riemann integral over (0, ∞), allowing that the integral can be +∞. The Lebesgue integral can then be defined by \[\int f\,d\mu = \int_0^\infty F(y)\,dy\] where the integral on the right is an ordinary improper Riemann integral, of a non-negative function (interpreted appropriately as +∞ if F(y) = +∞ on a neighborhood of 0).

Most textbooks, however, emphasize the simple functions viewpoint, because it is then more straightforward to prove the basic theorems about the Lebesgue integral.

Measure theory was initially created to provide a useful abstraction 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 later set theory developments showed (see non-measurable set), it is actually impossible to assign a length to all subsets of R in a way that preserves some natural additivity and translation invariance properties. This suggests that picking out a suitable class of measurable subsets is an essential prerequisite.

The Riemann integral uses the notion of length explicitly. 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 a certain class X of subsets of a set E, which satisfies a certain list of properties. These properties can be shown to hold in many different cases.

We start with a measure space (E, X, μ) where E is a set, X is a σ-algebra of subsets of E, and μ is a (non-negative) measure on E defined on the sets of X.

For example, E can be Euclidean n-space R or some Lebesgue measurable subset of it, X is the σ-algebra of all Lebesgue measurable subsets of E, and μ is the Lebesgue measure. In the mathematical theory of probability, we confine our study to a probability measure μ, which satisfies μ(E) = 1.

Lebesgue's theory defines integrals for a class of functions called measurable functions. A real-valued function f on E is measurable if the pre-image of every interval of the form (t, ∞) is in X: \[\{x\,\mid\,f(x) > t\} \in X\quad \forall t\in\mathbb{R}.\]

We can show that this is equivalent to requiring that the pre-image of any Borel subset of R be in X. The set of measurable functions is closed under algebraic operations, but more importantly it is closed under various kinds of point-wise sequential limits: \[\sup_{k \in \N} f_k, \quad \liminf_{k \in \N} f_k, \quad \limsup_{k \in \N} f_k\] are measurable if the original sequence (f_k), where k ∈ N, consists of measurable functions.

There are several approaches for defining an integral for measurable real-valued functions f defined on E, and several notations are used to denote such an integral. \[\int_E f \,d\mu = \int_E f(x)\,d\mu(x) = \int_E f(x)\,\mu(dx).\]

Following the identification in Distribution theory of measures with distributions of order 0, or with Radon measures, one can also use a dual pair notation and write the integral with respect to μ in the form \[\langle \mu, f\rangle.\]

The theory of the Lebesgue integral requires a theory of measurable sets and measures on these sets, as well as a theory of measurable functions and integrals on these functions.

One approach to constructing the Lebesgue integral is to make use of so-called simple functions: finite, real linear combinations of indicator functions. Simple functions that lie directly underneath a given function f can be constructed by partitioning the range of f into a finite number of layers. The intersection of the graph of f with a layer identifies a set of intervals in the domain of f, which, taken together, is defined to be the preimage of the lower bound of that layer, under the simple function. In this way, the partitioning of the range of f implies a partitioning of its domain. The integral of a simple function is found by summing, over these (not necessarily connected) subsets of the domain, the product of the measure of the subset and its image under the simple function (the lower bound of the corresponding layer); intuitively, this product is the sum of the areas of all bars of the same height. The integral of a non-negative general measurable function is then defined as an appropriate supremum of approximations by simple functions, and the integral of a (not necessarily positive) measurable function is the difference of two integrals of non-negative measurable functions.

Assuming that f is measurable and non-negative, the function \[f^*(t)\ \stackrel{\text{def}}{=}\ \mu \left (\{x \in E \mid f(x)>t\} \right ).\] is monotonically non-increasing. The Lebesgue integral may then be defined as the improper Riemann integral of f: \[\int_E f\,d\mu\ \stackrel{\text{def}}{=}\ \int_0^\infty f^*(t)\,dt.\] This integral is improper at the upper limit of ∞, and possibly also at zero. It exists, with the allowance that it may be infinite.

As above, the integral of a Lebesgue integrable (not necessarily non-negative) function is defined by subtracting the integral of its positive and negative parts.

Complex-valued functions can be similarly integrated, by considering the real part and the imaginary part separately.

If h = f + ig for real-valued integrable functions f, g, then the integral of h is defined by \[\int h \, d\mu = \int f \, d\mu + i \int g \, d\mu.\]

The function is Lebesgue integrable if and only if its absolute value is Lebesgue integrable (see Absolutely integrable function).

Consider the indicator function of the rational numbers, 1_Q, also known as the Dirichlet function. This function is nowhere continuous.

• \(1_{\mathbf Q}\) is not Riemann-integrable on [0, 1]: No matter how the set [0, 1] is partitioned into subintervals, each partition contains at least one rational and at least one irrational number, because rationals and irrationals are both dense in the reals. Thus the upper Darboux sums are all one, and the lower Darboux sums are all zero.
• \(1_{\mathbf Q}\) is Lebesgue-integrable on [0, 1] using the Lebesgue measure: Indeed, it is the indicator function of the rationals so by definition \[\int_{[0,1]} 1_{\mathbf Q} \,d\mu = \mu(\mathbf{Q} \cap [0,1]) = 0,\] because Q is countable.

A technical issue in Lebesgue integration is that the domain of integration is defined as a set (a subset of a measure space), with no notion of orientation. In elementary calculus, one defines integration with respect to an orientation: \[\int_b^a f = - \int_a^b f.\] Generalizing this to higher dimensions yields integration of differential forms on manifolds, and to Stokes' theorem as the generalization of the fundamental theorem of calculus. By contrast, Lebesgue integration provides an alternative generalization, integrating over subsets with respect to a measure; this can be notated as \[\int_A f\,d\mu = \int_{[a,b]} f\,d\mu\] to indicate integration over a subset A. For details on the relation between these generalizations, see Differential form § Relation with measures. The main theory linking these ideas is that of homological integration (sometimes called geometric integration theory), pioneered by Georges de Rham and Hassler Whitney.

With the advent of Fourier series, many analytical problems involving integrals arose whose satisfactory solution required interchanging limit processes and integral signs. However, the conditions under which the integrals \[\sum_k \int f_k(x) \,dx \quad \text{and} \quad \int \left[\sum_k f_k(x) \right] dx\] are equal proved quite elusive in the Riemann framework. There are some other technical difficulties with the Riemann integral. These are linked with the limit-taking difficulty discussed above.

As shown above, the indicator function 1_Q on the rationals is not Riemann integrable. In particular, the Monotone convergence theorem fails. To see why, let {a_k} be an enumeration of all the rational numbers in [0, 1] (they are countable so this can be done). Then let \[g_k(x) = \begin{cases} 1 & \text{if } x = a_j, j\leq k \\ 0 & \text{otherwise} \end{cases}\]

The function g_k is zero everywhere, except on a finite set of points. Hence its Riemann integral is zero. Each g_k is non-negative, and this sequence of functions is monotonically increasing, but its limit as k → ∞ is 1_Q, which is not Riemann integrable.