Lebesgue Integration Introduction Construction Of The Lebesgue

The Lebesgue integral plays an important role in the branch of mathematics called real analysis and several other fields.

The Lebesgue integral is named for Henri Lebesgue ( 1875- 1941). The pronunciation of his name may be approximated as leh BEG.

1 Introduction

The integral of a function f can be interpreted as the area S 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 formalism in 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 integrals 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, the behavior of the Riemann integral in limit processes is 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 integrals 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 function which is 0 where its argument is irrational and 1 otherwise has a Lebesgue integral, but it does not have a Riemann integral.

We now give a highly technical description. It is possible to skip directly to the discussion heading for further technical and historical justification of the Lebesgue integral if the reader is so inclined.

2 Construction of the Lebesgue integral

Let μ be a (non-negative) measureMeasure theory In mathematics, a measure is a function that assigns a number, e. a "size", "volume", or "probability", to subsets of a given set. The concept is important in mathematical analysis and probability theory. Measure theory is that branch of re on a sigma-algebraIn mathematics, a sigma;-algebra (or sigma;-field X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S''. The concept is important in mathematical anal X over a set E. (In real analysis, E will typically be Euclidean n-spaceEuclidean space is the usual n dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid. Formally, for any non-negative integer n n dimensional Euclidean space is the set R n (where R is the set of real numbers Rⁿ or some Lebesgue measurableMeasure theory The Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration. Sets which can be assigned a volume are called Lebesgue measura subset of it, X will be the sigma-algebra of all Lebesgue measurable subsets of E, and μ will be the Lebesgue measure. In probability and statistics, μ will be a probabilityThe word probability derives from the Latin probare (to prove, or to test). Informally, probable is one of several words applied to uncertain events or knowledge, being more or less interchangeable with likely risky hazardous uncertain and doubtful depend measure on a probability space E.) We build up an integral for real-valued functions defined on E as follows.

Fix a set S in X (S ∈ X, S ⊂ E) and let f be the function on E whose value is 0 outside of S and 1 inside of S (i.e., f(x) = 1 if x is in S, otherwise f(x) = 0.) This is called the indicator functionThis article is about the characteristic function in set theory. For characteristic function in probability theory see characteristic function In mathematical subfield of set theory, the indicator function or characteristic function is a function defined or characteristic function of S and is denoted 1_S.

To assign a value to ∫1_S consistent with the given measure μ, the only reasonable choice is to set:

We extend by linearity to the linear spanIn the mathematical subfield of linear algebra, the linear span of a set of vectors is the set of all linear combinations of the vectors. The linear span of a set of vectors is a therefore a vector space but unlike a basis the vectors need not be linearly of indicator functions:

where the sum is finite and the coefficients a_k are real numbers. Such a finite linear combinationIn mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalisations given at the end of of indicator functions is called a simple function. Note that a simple function can be written in many ways as a linear combination of characteristic functions, but the integral will always be the same.

Now the difficulties begin as we attempt to take limits so that we can integrate more general functions. It turns out that the following process works and is most fruitful.

Let f be a non-negative function supported on the set E (we allow it to attain the value +∞, in other words, f takes values in the extended real number line.) We define ∫f to be the supremum of ∫s where s varies over all simple functions which are under f (that is, s(x) ≤ f(x) for all x.) This is analogous to the lower sums of Riemann. However, we will not build an upper sum, and this fact is important in getting a more general class of integrable functions. One can be more explicit and mention the measure and domain of integration:

There is the question of whether this definition makes sense (do simple function or indicating function keep the same integral?) There is also the question of whether this corresponds in any way to a Riemann notion of integration. It is not so hard to prove that the answer to both questions is yes.

We have defined ∫f for any non-negative function on E; however for some functions ∫f will be infinite. Furthermore, desirable additive and limit properties of the integral are not satisfied, unless we require that all our functions are measurable, meaning that the pre-image of any interval is in X. We will make this assumption from now on.

To handle signed functions, we need a few more definitions. If f is a function of the measurable set E to the reals (including ± ∞), then we can write f = g − h where g(x) = (f(x) if f(x)>0, 0 otherwise) and h(x) = (−f(x) if f(x) < 0, 0 otherwise). Note that both g and h are non-negative functions. Also note that |f| = g + h. If ∫|f| is finite, then f is called Lebesgue integrable. In this case, both ∫g and ∫h are finite, and it makes sense to define ∫f by ∫g − ∫h. It turns out that this definition is the correct one. Complex valued functions can be similarly integrated, by considering the real part and the imaginary part separately.

2.1 Intuitive interpretation

Let us imagine that it is desired to find a mountain's volume (above sea level) and that the mountain's boundaries are marked out clearly (they are the boundaries of integration). So in this case we have a double-integral.

The Riemann approach to integrating the mountain's volume can be imagined thus: cut the mountain up into vertical slices, each one having a square base at sea-level. Pick a representative point in this square, then find the point at the top of the slice such that a plumb line dropped from this point at the top falls directly on the representative point at the base. The height of the plumb line is the representative height of the slice, and the representative volume is given by multiplying the representative height by the area of the square. The Riemann sum is the sum of the representative volumes of all the slices, and the Riemann integral is the limit of the Riemann sums as each slice is cut into four sub-slices, each subslice having one-fourth base area of the slice, so that the sum is recomputed for the newer set of subslices, then the slices are cut up again, and then their sum recomputed, and so on.

The Lebesgue approach can be imagined thus: draw a contour map of the mountain. Find the contour (or set of contours) with the lowest height. Find the total area enclosed (within the map) by this set of contours, i.e. find the " measure" of this set of contours. Multiply this measure by the height represented by the set of contours: the product will be one summand of a "Lebesgue sum".

Then find a contour, or set of contours, which are at one step higher in height (lowest height of the remaining contours). Calculate the measure of the area enclosed by them. Multiply the measure by the difference in height (from the previous step), and the product will be another summand of the "Lebesgue sum".

Repeat this process for successive higher levels of contours, until the highest set of contours has been processed. The resulting sum is the linear span: each contour corresponding to an indicator function.

The sum can be refined by adding intermediate contours to the map: halving the difference between successive heights, then recomputing the sum.

If Riemann sums are like raster graphics, or like pointillism, then Lebesgue sums are like vector graphics, or like contour-bounded solid blocks of color.

2.2 Example

Consider the indicator function of the rational numbers, I_Q, which is nowhere continuous:

What is

equal to?

This function is Darboux-nonintegrable (and therefore Riemann-nonintegrable). No matter how the set [0,1] is partitioned, each partition will contain at least one rational and at least one irrational number, since they are both infinitely dense in the reals. Then, the representative "height" of each "slice" will always range from a minimum of zero to a maximum of one, so the upper Darboux integral will be one, and the lower Darboux integral will be zero, and so the function is Darboux-nonintegrable.

But this function is Lebesgue-integrable. The function has a pair of "heights": 0 and 1. The set has height 1, and the measure of this set, , is zero, because has a countable number of elements.

Due to additivity of measure,

but the height of set is 0, so

3 Properties of the Lebesgue integral

Every reasonable notion of integral needs to be linear and monotone, and the Lebesgue integral is: if f and g are integrable functions and a and b are real numbers, then af + bg is integrable and ∫(af + bg) = a∫f + b∫g; if f ≤ g, then ∫f ≤ ∫g.

Two functions which only differ on a set of μ-measure zero have the same integral, or more precisely: if μ({x : f(x) ≠ g(x)}) = 0, then f is integrable if and only if g is, and in this case ∫ f = ∫ g.

One of the most important advantages that the Lebesgue integral carries over the Riemann integral is the ease with which we can perform limit processes. Three theorems are key here.

The monotone convergence theorem states that if f_k is a sequence of non-negative measurable functions such that f_k(x) ≤ f_k+1(x) for all k, and if f = lim f_k, then ∫f_k converges to ∫f as k goes to infinity. (Note: ∫f may be infinite here.)

Fatou's lemma states that if f_k is a sequence of non-negative measurable functions and if f = liminf f_k, then ∫f ≤ liminf ∫f_k. (Again, ∫f may be infinite.)

The dominated convergence theorem states that if f_k is a sequence of measurable functions with pointwise limit f, and if there is an integrable function g such that |f_k| ≤ g for all k, then f is integrable and ∫f_k converges to ∫f.

4 Equivalent formulations

If f is non-negative, then ∫f dμ is precisely the area under the curve as measured by the product measure μ × λ where λ is the Lebesgue measure for R.

One can also circumvent measure theory entirely. The Riemann integral exists for any continuous function f of compact support. Then we use functional analysis to obtain the integral for more general functions. Let C_c be the space of all real-valued compactly supported continuous functions of R. Define a norm on C_c by

||f|| = ∫ |f(x)|

Then C_c is a normed vector space (and in particular, it is a metric space.) All metric spaces have completions, so let L¹ be its completion. This space is isomorphic to the space of Lebesgue integrable functions (modulo sets of measure zero). Furthermore, the Riemann integral ∫ defines a continuous functional on C_c which is dense in L¹ hence ∫ has a unique extension to all of L¹. This integral is precisely the Lebesgue integral.

In this formulation, the limit taking theorems are hard to prove. However, in more general cases (such as when the functions, or perhaps the measures, take values in a large vector space instead of Rⁿ) this approach is a fast way of obtaining an integral.

5 Discussion

Here we discuss the limitations of the Riemann integral and the greater scope offered by the Lebesgue integral. We presume a working understanding of the Riemann integral.

With the advent of Fourier series, there arose the need to exchange summation and integral signs much more often. However, the conditions under which ∑_k∫f_k and ∫∑_kf_k are equal proved quite elusive in the Riemann framework. It may come as a surprise to the casual reader that these two quantities may not be equal, so an example helps:

Let f_k(x) be 1 on (k, k+1] and -1 on (k+1, k+2] and 0 everywhere else.
Then, ∑_k=1^∞f_k(x) = f(x) where f(x) is zero everywhere except on (1, 2] where it is 1. Hence, ∫∑f_k = ∫f = 1.
However, ∫f_k = 0 for every k, hence ∑∫f_k = 0.

However, it was clear from experience that in many very useful situations, the sum and the integral did commute. It was very important to be able to describe which conditions enabled the exchange of the sum and integral signs. Unfortunately, the Riemann integral is poorly equipped to deal with this question; its main useful convergence theorem being the uniform convergence theorem: if f_k are Riemann-integrable functions of [a, b] converging uniformly to f, then ∫f_k converges to ∫f. Since Fourier series rarely converge uniformly, this theorem is clearly insufficient.

There are some other technical difficulties with the Riemann integral. These are linked with the limit taking difficulty discussed above.

If H(x) is a function of [0,1] which is 0 everywhere, except that it is 1 on the rational numbers (see nowhere continuous), then it is not Riemann integrable. This is because, in the calculation of its upper sum, any rectangle used will have height 1 (because all rectangles contain rational points) and in the lower sum, any rectangle used will have height 0 (because all rectangles contain irrational points.) Hence the lower sum is 0 and the upper sum is 1.

This means that the monotone convergence theorem does not hold. The monotone convergence theorem would say that if f_k(x) is a sequence of non-negative functions increasing monotonically in k to f(x), then the integrals of ∫ f_k(x) dx should converge to ∫ f(x) dx. To see why this is so, 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 be the function which is 1 on a_k and 0 everywhere else. Lastly let f_k = g₁ + g₂ + ... + g_k. Then f_k is zero everywhere except on a finite set of points, hence its Riemann integral is zero. The sequence f_k is also clearly non-negative and monotonously increasing to H(x), but H(x) isn't Riemann integrable.

The Riemann integral can only integrate functions on an interval. The simplest extension is to define ∫_{− ∞}^∞f(x) dx by the limit of ∫_−a^af(x) dx as a goes to +∞. However, this breaks translation invariance: if f and g are zero outside some interval [a, b] and are Riemann integrable, and if f(x) = g(x + y) for some y, then ∫ f = ∫ g. However, with this definition of the improper integral (this definition is sometimes called the improper Cauchy principal value about zero), the functions f(x) = (1 if x > 0, −1 otherwise) and g(x) = (1 if x > 1, −1 otherwise) are translations of one another, but their improper integrals are different. (∫ f = 0 but ∫ g = − 2.)

5.1 Towards a better integration theory

The solution, as it turns out, is to study an even simpler problem first. The observation is that, if we have a notion of length, we can turn it into a notion of area. Instead of measuring the area of a surface in the plane, we turn our attention to measuring the length of subsets of the real line. One obvious requirement is that an interval [a, b] should have a length of b-a. What other demands we should put on the notion of length is less clear, and much effort was put into obtaining a useful definition. In fact, the term length was first used, but its construction was misguided, and a later, more useful construction is in use today; it is called the measure.

Measure theory enables us to calculate the lengths of subsets of the real line. It also fully classifies which sets have a length, and which sets do not have a reasonable notion of length. By spending the extra effort into calculating lengths carefully, we now have a more solid foundation to work with.

Of course, the Riemann integral uses the notion of length anonymously. Indeed, the element of calculation for the Riemann integral is the rectangle [a, b] × [c, d], whose area is calculated to be (b-a)(c-d). Obviously the numbers b-a and c-d are meant to be the lengths of [a, b] and [c, d]. However, we can now augment the Riemann integral. Indeed, Riemann could only use rectangles because he could only measure intervals. Equipped with a measure μ, we can calculate the length of sets much more interesting than intervals. So, if X and Y are μ-measurable, we can easily define the area of the cartesian product X × Y to be μ(X) μ(Y). This definition clearly generalizes Riemann's notion of area of a rectangle. In the context of Lebesgue integration, sets such as X × Y are sometimes called rectangles, even though they are far more complicated than the quadrilaterals of the same name.

With the ability to measure the area of more complex rectangles, we can attempt to integrate more complex functions. One crucial, but nonobvious step, was to drop the notion of upper sum. While upper sums work just fine for bounded functions of bounded intervals, there is a clear problem for unbounded functions, or functions which are supported by all of the real line. For instance, the function f(x) = 1/x² for x > 1 would necessarily have an infinite upper sum, however it can be shown that this function has a finite integral.

Dropping the upper sum robs us of our main way of checking for integrability of functions. It isn't obvious how to decide on the integrability of functions while maintaining a consistent theory. It is very fortunate that a simple (if technical) definition is available.

The resulting theory of integration is much more accurate in describing limit taking processes. Many of the original questions posed by Fourier series (about swapping the integral and summation signs) are answerable using one or another of the various Lebesgue integral limit theorems (the main ones are monotone convergence, dominated convergence and Fatou's lemma; see above.)

6 See also

Calculus

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