Lebesgue integration

The Lebesgue integral ( after Henri Léon Lebesgue ) is the integral concept of modern mathematics, which enables the integration of functions that are defined on arbitrary measure spaces. In the case of real numbers with the Lebesgue measure is the Lebesgue integral of a real generalization of the Riemann integral dar.

Intuitively, this means: To approach the Riemann integral (blue) is the x-axis is divided into intervals (partitions) and rectangles according to the function value at a reference point within the corresponding intervals constructed and these faces are added. On the other hand (red) is the approximation of the Lebesgue integral divides the ordinate axis into intervals and the faces for the approximation resulting from a support point of the respective Ordinatenintervalls multiplied by the total length of the union of the inverse images of the Ordinatenintervalls (same reds ). The sum of the areas thus formed gives an approximation of the Lebesgue integral. The total length of the pre-image set is referred to as its measure. Compare this with the quotation by Henri Lebesgue in the lowest section of this article.

Is as a Riemann integral defined by the convergence of the area of ​​a series of step functions, the Lebesgue integral defined by the convergence of a sequence of so-called simple functions.

History to the Lebesgue integral

The reasons for the differential and integral calculus begins in the 17th century with Isaac Newton and Gottfried Wilhelm Leibniz ( 1687 appears Newton " Philosophiae Naturalis Principia Mathematica "). It represents a milestone in the history of science, you had but now for the first time a mathematical concept for the description of continuous, dynamic processes in nature and - motivated - to calculate curvilinear surfaces with boundary deterministic. However, it should be for many decades to pass before the integral calculus was made in the mid 19th century by Augustin Louis Cauchy and Bernhard Riemann on a solid theoretical foundation.

The generalization of the so-called Riemann integral to higher-dimensional spaces, for example, to calculate the volumes of arbitrary bodies in space, however, proved difficult. The development of a modern and more efficient integral term is inextricably linked to the development of measure theory. In fact, mathematicians began to examine systematically until very late how arbitrary subsets of the can assign a volume in a meaningful way. Indispensable for this work was the strict axiomatic justification of the real numbers by Richard Dedekind and Georg Cantor and the justification of set theory by Cantor, the end of the 19th century.

Getting answers to the question of the volume of individual subsets of tasks, for example, Giuseppe Peano and Marie Ennemond Camille Jordan. However, a satisfactory solution to this problem was achieved only Émile Borel and Henri Lebesgue by the construction of Lebesgue measure. 1902 Lebesgue formulated in his Paris Thèse for the first time the modern measurement problem and explicitly pointed out, it can not solve in full generality, but only for a very specific class of sets, which he called measurable sets. In fact, it turns out that the measurement problem is not generally solvable, ie actually exist quantities which one can not assign a meaningful measure ( see Theorem of Vitali, Banach - Tarski paradox ). By the construction of the Lebesgue measure of the way for a new, generalizable integral term is now stood open. The first definition of the Lebesgue integral was because Henri Lebesgue in his Thèse equal to itself Other important definitions of the Lebesgue integral came a little later by William Henry Young ( 1905) and Frigyes Riesz (1910). The definition presented below, which in the literature is now the most common, followed by the construction Youngs.

Nowadays, the Lebesgue integral, the integral concept of modern mathematics. Its generalizability and his - from a mathematical perspective - beautiful properties also make it an indispensable tool in the functional analysis, physics and probability theory.

To construct the Lebesgue integral

Measure space and measurable quantities

The Lebesgue integral is defined for functions on an arbitrary measure space. Put simply, a measure space is a set Ω with an additional structure that allows to associate specific subsets of a measure, such as their geometric length (or its volume ). The measure, which does this is called Lebesgue measure. A subset of which you can assign a measure is, measurable. If A is a measurable quantity, is denoted by the measure of. The measure of a measurable set is a non-negative real number or. For the Lebesgue measure of a subset of the one writes instead usually.

Integration of simple functions

Just as the Riemann integral is constructed by means of approximation by step functions, we construct the Lebesgue integral with the help of so-called simple functions. A simple function, also called elementary function is a non-negative measurable function that takes only finitely many function values ​​? I. Thus, any simple function can be written as

It must be a positive real number, the ( measurable ) set on which the function takes on the value and the characteristic function.

Now can be a very natural way the integral of a simple function define:

The integral of over is thus simply the sum of the products of function and value of measure of the set on which the function assumes the value.

Integration of non-negative functions

Now one first defines the integral for non-negative functions, ie, functions that accept negative values ​​. Condition for the integrability of a function is its measurability.

A non - negative function, B Borel σ -algebra if and only measurable if there exists a sequence of simple functions that converges pointwise and monotonically to f. Now we define the integral of a non-negative measurable function by

Which are simple and pointwise and monotonically converge to f. The limit is independent of the particular choice of the sequence. The integral can also assume the value.

Frequently one finds in the literature the following equivalent definition:

So you define the integral of a non-negative measurable function by approximating the function "from below" arbitrary accuracy by simple functions.

Integration of arbitrary measurable functions and integration

To define the integral any measurable function, this is broken down into its positive and negative component, the two integrated separately from one another and pulls the integrals. But that only makes sense if the values ​​of these two integrals are finite (at least the value of one of the two integrals).

The positive part of a function f is ( pointwise ) is defined as.

The negative part is in accordance with ( pointwise ) given by.

Μ is a function of a quasi - integrated or integrated with respect to the quasi measure μ, if at least one of the two integrals

Is finite.

In this case, ie

The integral of over.

For all measurable subsets then

The integral of over.

A function is called μ - integrable or integrable with respect to the measure μ if both integrals

Finite. Equivalent to the condition

Obviously, any integrable function is quasi- integrable.

Important properties of the Lebesgue integral

The integral is linear in ( space of integrable functions ), ie for integrable functions and arbitrary and is also integrable and we have:

The integral is monotone, ie, and two measurable functions, it shall

The integral can be separated

Is measurable, it shall

Convergence Theorems

One of the main advantages of the Lebesgue integral are from a mathematical point of view very beautiful convergence theorems. This relates to the interchangeability of limit and integral function in sequences of the form. The main convergence theorems are:

• Is integrable,
• And

Null sets, and almost - anywhere existing properties

A lot of that has the degree 0 is called null set. In the case of the Lebesgue measure also specifically Lebesgue null set. Thus, if f is an integrable function with and, then:

Since the integral over the amount of N zero assumes a value of 0. ( Denotes the set without the quantity)

Consequently, the value of the integral does not change if you change the function f on a null set. Has a function a property ( continuity, pointwise convergence, etc.) on the entire domain except for a set of measure 0, it is said, this property consists almost - everywhere. In the Lebesgue integration theory, it is therefore often useful two functions match almost - everywhere, also to be regarded as equal - one combines them an equivalence class together (see also Lp).

It is even often so that one functions that are only defined almost everywhere ( for example, the pointwise limit of a sequence of functions which converges almost everywhere only ), as functions conceives on the entire room and without hesitation

Writes, even if f is not defined at all. This approach is justified by the fact that any extension of f only on a null set N of f is different and thus the integral over the continuation of the very same value as the integral over.

This convention allows many sets easier to formulate, for example, one might set of the dominated convergence (see above) also write as:

Be measurable, integrable, is convergent almost everywhere and each is limited almost everywhere. Then every limit and the integrable and we have:

One must note that a null set only in the sense of measure is negligible " small". But it can also contain quite infinitely many elements. For example, the amount, ie, the set of rational numbers as a subset of the real numbers a Lebesgue -null set. The Dirichlet function

So in the above sense is equal to the function, the constant assumes the value zero (zero function), although there is no matter how small environment in which match their values. A known uncountable ( to the same powerful ) Lebesgue -null set is the Cantor set.

Spellings

For the Lebesgue integral number of conventions are used: The following is a measurable quantity. If you want to specify the integration variable in the integration, one writes

If the Lebesgue measure, we write instead of just the one-dimensional case, we also write

For the integral over the interval or.

If the measure has a Radon Nikodým - density with respect to the Lebesgue measure applies

In applications, the notation

Often used when formally has no density. However, this is only useful if one perceives not as a function but as a distribution.

Is the measure defined by a cumulative function of the case, it also writes

( Stieltjes integral).

Is a probability measure, then one writes for

( Expected value ). In theoretical physics, the notation is used in the functional analysis sometimes the spelling.

Riemann and Lebesgue integral

In the case with the Lebesgue measure applies: If a function on a compact interval Riemann - integrable, then it is also Lebesgue integrable and the values ​​of both integrals coincide. On the other hand, not every Lebesgue integrable function also Riemann integrable. But has a function at most countably many points of discontinuity, as also follows from Lebesgue integrability the Riemann integrability.

On the other hand must have a improperly Riemann integrable function is not Lebesgue integrable as a whole, the corresponding limit of Lebesgue integrals exist, however, by the above remarks and provides the same value as for the Riemann integrals. However, if improperly Riemann integrable, then it is Lebesgue integrable even as a whole.

It is easy to provide an example of an improper Riemann - integrable function which is not Lebesgue integrable: Indeed, if f is a step function with the surfaces 1, -1 / 2, 1/3, etc., then f is improper Riemann - integrable. Because the integral corresponds precisely to the alternating harmonic series. If f Lebesgue integrable, then would apply. However, this is not the case, as the harmonic number is divergent. Consequently, the corresponding Lebesgue integral does not exist. The situation is shown in the following image:

More important is the reverse case of a Lebesgue integrable function that is not Riemann integrable.

The best known example is the Dirichlet function:

F is not Riemann integrable, since all lower sums always 0 and all upper sums are always 1. However, since the set of rational numbers, the set of real numbers is a Lebesgue -null set, the function almost everywhere 0 So does the Lebesgue integral and has the value 0

The main difference in the approach to the integration by Riemann or Lebesgue is that the Riemann integral to the definition range ( x-axis ), the Lebesgue integral, however, the image rate ( ordinate) is divided to the function. In the above examples can already see that this difference may well prove decisive.

"You can say that you look at the procedure of Riemann behaves like a merchant without a system of coins and banknotes mentioned in order, as he gets into his hand; while we proceed as a prudent businessman, who says:

Etc., so I have total. The two methods lead sure the merchant to the same result, because he - how rich he may be - has to count only a finite number of banknotes; but for us, we have to add an infinite number of Indivisiblen, the difference between the two procedures is essential. "

Bochner integral

A direct generalization of the Lebesgue integral for Banach space - valued functions is the Bochner integral dar. It inherits almost all properties of the Lebesgue integral, such as the set of the dominated convergence.