Lebesgue integral

This article deals with the general term integral, for the special integration with regard to the Lebesgue measure , see there.

Figure 1: Illustration of the limit value formation with the Riemann integral (blue) and the Lebesgue integral (red)

The Lebesgue integral (after Henri Léon Lebesgue [ ɑ̃ʁiː leɔ̃ ləˈbɛg ]) is the integral term in modern mathematics, which enables the integration of functions that are defined on any dimensional space. In the case of real numbers with the Lebesgue measure , the Lebesgue integral represents a real generalization of the Riemann integral .

In clear terms, this means: To approximate the Riemann integral (Fig. 1 blue), the abscissa axis is subdivided into intervals ( partitions ) and rectangles are constructed according to the function value at a support point within the relevant intervals and these areas are added. On the other hand, to approximate the Lebesgue integral (Fig. 1 red) the ordinate axis is divided into intervals and the areas for approximation result from a support point of the respective ordinate interval multiplied by the total length of the union of the archetypes of the ordinate interval (same red tones). The sum of the areas formed in this way gives an approximation of the Lebesgue integral. The total length of the archetype set is also called its size . Compare the quote from Henri Lebesgue in the bottom section of this article.

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

History of the Lebesgue integral

The foundation of differential and integral calculus begins in the 17th century with Isaac Newton and Gottfried Wilhelm Leibniz (Newton's “Philosophiae Naturalis Principia Mathematica” appeared in 1687). It represents a milestone in the history of science, as a mathematical concept was now available for the first time to describe continuous, dynamic processes in nature and - motivated by this - to calculate surfaces with curvilinear borders. However, many decades would pass before the integral calculus was put on a solid theoretical foundation by Augustin Louis Cauchy and Bernhard Riemann in the middle of the 19th century .

The generalization of the so-called Riemann integral to higher-dimensional spaces, for example to calculate the volumes of arbitrary bodies in space, turned out to be difficult. The development of a more modern and more powerful integral concept is inseparably linked with the development of the theory of measure . In fact, mathematicians began to systematically investigate how any subsets of the can be assigned to a volume in a meaningful way. An indispensable prerequisite 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 towards the end of the 19th century. ${\displaystyle \mathbb {R} ^{n}}$

Giuseppe Peano and Marie Ennemond Camille Jordan , for example, gave initial answers to the question of the volume of any subsets . A satisfactory solution to this problem was only found by Émile Borel and Henri Lebesgue by constructing the Lebesgue measure. In 1902 Lebesgue formulated the modern measurement problem for the first time in his Paris Thèse and explicitly pointed out that it could not be solved in full generality, but only for a very specific class of sets, which he called measurable sets . In fact, it should turn out that the measure problem cannot be solved in general, ie sets actually exist to which no meaningful measure can be assigned (see Vitali theorem , Banach-Tarski paradox ). The construction of the Lebesgue measure now opened the way for a new, generalizable integral term. The first definition of the Lebesgue integral was given by Henri Lebesgue himself in his Thèse . Further important definitions of the Lebesgue integral came a little later from William Henry Young (1905) and Frigyes Riesz (1910). The definition presented below, which is now the most common in the specialist literature, follows the construction of Young. ${\displaystyle \mathbb {R} ^{n}}$

Nowadays the Lebesgue integral is the integral term in modern mathematics. Its generalizability and its - from a mathematical point of view - beautiful properties also make it an indispensable tool in functional analysis , physics and probability theory .

For the construction of the Lebesgue integral

Measure space and measurable quantities

The Lebesgue integral is defined for functions on any dimension space . Put simply, a measure space is a set with an additional structure that allows a measure to be assigned to certain subsets , e.g. B. their geometric length (or their volume). The measure that achieves this is called the Lebesgue measure . A subset of to which a measure can be assigned is called measurable. If there is a measurable amount, then one denotes the measure of . The measure of a measurable quantity is a nonnegative real number or . For the Lebesgue measure of a subset of the one writes instead, usually . ${\displaystyle \Omega }$ ${\displaystyle A}$ ${\displaystyle \Omega }$ ${\displaystyle A}$ ${\displaystyle {\mathcal {\mu }}(A)}$ ${\displaystyle A}$ ${\displaystyle +\infty }$ ${\displaystyle A}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle {\mathcal {\lambda }}(A)}$

Integration of simple functions

Just as the Riemann integral is constructed using approximation using staircase functions , the Lebesgue integral is constructed using so-called simple functions . This practice is sometimes referred to as "algebraic induction" and is used in much of the evidence for measurable functions. A simple function, also called an elementary function, is a non-negative measurable function that only takes on a finite number of function values. So any simple function can be written as ${\displaystyle \alpha _{i}}$ ${\displaystyle \phi }$

${\displaystyle \phi =\sum _{i=1}^{n}\alpha _{i}\chi _{A_{i}}}$.

Here, a positive real number , the (measurable) amount on which the function takes the value , and the characteristic function are to . ${\displaystyle \alpha _{i}}$ ${\displaystyle A_{i}}$ ${\displaystyle \alpha _{i}}$ ${\displaystyle \chi _{A_{i}}}$ ${\displaystyle A_{i}}$

Now the integral of a simple function can be defined in a very natural way : ${\displaystyle \phi }$

${\displaystyle \int _{\Omega }\phi \ \mathrm {d} \mu =\sum _{i=1}^{n}\alpha _{i}\mu (A_{i})}$

The integral of over is therefore simply the sum of the products of the function value of and the measure of the amount to which the function takes the respective value. ${\displaystyle \phi }$ ${\displaystyle \Omega }$ ${\displaystyle \phi }$

Integration of non-negative functions

Now we first define the integral for non-negative functions, i. H. for functions that do not assume negative values. The precondition for the integrability of a function is that it can be measured .

A non-negative function , Borel σ-algebra is exactly measurable when there is a sequence are of simple functions pointwise and monotonically increasing to converge . The integral of a non-negative, measurable function is now defined by ${\displaystyle f\colon \left(\Omega ,\Sigma ,\mu \right)\rightarrow ({\overline {\mathbb {R} }},{\mathcal {B}})}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f}$

${\displaystyle \int _{\Omega }f\,\mathrm {d} \mu =\lim _{n\rightarrow \infty }\int _{\Omega }f_{n}\,\mathrm {d} \mu }$,

where they are simple and converge pointwise and monotonically increasing towards . The Limes is independent of the particular choice of the sequence . The integral can also take the value . ${\displaystyle f_{n}}$ ${\displaystyle f}$ ${\displaystyle f_{n}}$ ${\displaystyle +\infty }$

The following equivalent definition is often found in the literature:

${\displaystyle \int _{\Omega }f\,\mathrm {d} \mu =\sup \left\lbrace \int _{\Omega }\phi \,\mathrm {d} \mu \,\vert \,\phi \ {\text{einfach}},0\leq \phi \leq f\right\rbrace }$

The integral of a non-negative measurable function is thus defined by approximating the function “from below” with arbitrary precision using simple functions.

Integration of any measurable functions and integrability

To define the integral of any measurable function, one breaks it down into its positive and negative part , integrates these two individually and subtracts the integrals from each other. But this 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 is (pointwise) defined as . ${\displaystyle f^{+}}$ ${\displaystyle f}$ ${\displaystyle f^{+}=\max\{f,0\}}$

The negative part is defined accordingly (point by point) by . ${\displaystyle f^{-}}$ ${\displaystyle f^{-}=\max\{-f,0\}}$

It then (point-wise) , , and . ${\displaystyle f^{+}\geq 0}$ ${\displaystyle f^{-}\geq 0}$ ${\displaystyle f=f^{+}-f^{-}}$ ${\displaystyle f^{+}+f^{-}=|f|}$

A function is called µ-quasi-integrable or quasi-integrable with respect to the measure µ , if at least one of the two integrals

${\displaystyle \int _{\Omega }f^{+}\ \mathrm {d} \mu }$ or ${\displaystyle \displaystyle \int _{\Omega }f^{-}\ \mathrm {d} \mu }$

is finite.

In this case it means

${\displaystyle \int _{\Omega }f\ \mathrm {d} \mu =\int _{\Omega }f^{+}\ \mathrm {d} \mu -\int _{\Omega }f^{-}\mathrm {d} \mu }$.

the integral of over . ${\displaystyle \mu }$ ${\displaystyle f}$ ${\displaystyle \Omega }$

Then for all measurable subsets ${\displaystyle A\subseteq \Omega }$

${\displaystyle \int _{A}f\mathrm {d} \mu =\int _{\Omega }f\cdot \chi _{A}\ \mathrm {d} \mu }$

the integral of over . ${\displaystyle \mu }$ ${\displaystyle f}$ ${\displaystyle A}$

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

${\displaystyle \int _{\Omega }f^{+}\ \mathrm {d} \mu }$ and ${\displaystyle \displaystyle \int _{\Omega }f^{-}\ \mathrm {d} \mu }$

are finite. The condition is equivalent to this

${\displaystyle \int _{\Omega }|f|\,\mathrm {d} \mu <\infty }$.

Obviously every function that can be integrated is quasi-integrated.

Spellings

Numerous notations are used for the Lebesgue integral: Let the following be a measurable set. If you want to specify the integration variable during integration , you write ${\displaystyle A\subseteq \Omega }$ ${\displaystyle x}$

${\displaystyle \int _{A}f(x)\,\mathrm {d} \mu (x)}$or or also . ${\displaystyle \int _{A}f(x)\,\mu (\mathrm {d} x)}$ ${\displaystyle \int _{A}\mu (\mathrm {d} x)\,f(x)}$

If the Lebesgue measure is used, you write instead of simply , in the one-dimensional case you also write ${\displaystyle \mu }$ ${\displaystyle \mathrm {d} \mu (x)}$ ${\displaystyle \mathrm {d} x}$ ${\displaystyle \Omega =\mathbb {R} }$

${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x}$

for the integral over the interval or . ${\displaystyle [a,b]}$ ${\displaystyle ]a,b[}$

If the measure has a Radon-Nikodým density with respect to the Lebesgue measure, then applies ${\displaystyle \mu }$ ${\displaystyle h}$

${\displaystyle \int _{A}f(x)\,\mathrm {d} \mu (x)=\int _{A}f(x)\,h(x)\mathrm {d} x}$.

In application areas, the notation

${\displaystyle \int _{A}f(x)\,h(x)\,\mathrm {d} x}$

often also used when formally has no density. However, this only makes sense if one sees it as a distribution rather than a function . ${\displaystyle \mu }$ ${\displaystyle h}$

If the measure in the case is defined by a distribution function, one also writes ${\displaystyle \mu }$ ${\displaystyle \Omega =\mathbb {R} }$ ${\displaystyle F}$

${\displaystyle \int _{A}f(x)\,\mathrm {d} F(x)}$ or ${\displaystyle \int _{A}f\,\mathrm {d} F}$

( Lebesgue-Stieltjes integral ).

If a probability measure is , one also writes for ${\displaystyle \mu }$ ${\displaystyle \mathbb {E} (f)}$

${\displaystyle \int _{\Omega }f\,\mathrm {d} \mu \,}$

( Expected value ). In theoretical physics the notation is used, in functional analysis sometimes the notation . ${\displaystyle \langle f\rangle }$ ${\displaystyle \ \mu (f)\ }$

Zero quantities and properties that exist almost everywhere

A set that has the measure 0 is called a null set , in the case of the Lebesgue measure also specifically a Lebesgue null set . So if with and is an integrable function, then: ${\displaystyle N\subset \Omega }$ ${\displaystyle N\subset \Omega }$ ${\displaystyle \ \mu (N)=0\ }$ ${\displaystyle f}$

${\displaystyle \int _{\Omega }f\,\mathrm {d} \mu =\int _{\Omega \setminus N}f\,\mathrm {d} \mu +\int _{N}f\,\mathrm {d} \mu =\int _{\Omega \setminus N}f\,\mathrm {d} \mu }$

because the integral over the zero set takes the value 0. ( denotes the amount without the amount ) ${\displaystyle N}$ ${\displaystyle \Omega \setminus N}$ ${\displaystyle \Omega }$ ${\displaystyle N}$

Hence the value of the integral does not change if the function is changed on a zero set. If a function has a property (continuity, point-wise convergence, etc.) on the entire domain with the exception of a set of measure 0, this property is said to exist almost everywhere . In Lebesgue's integration theory, it therefore often makes sense to consider two functions that match almost everywhere as the same - they are combined to form an equivalence class (see also L ^p ). ${\displaystyle f}$

It is even often the case that functions that are only defined almost everywhere (e.g. the point-by-point limit of a function sequence that only converges almost everywhere) are understood as functions in the whole space and without hesitation

${\displaystyle \int _{\Omega }f\,d\mu }$

writes, even if it is not entirely defined at all . This procedure is justified by the fact that each continuation of itself differs only on a zero set of and thus the integral of the continuation over has exactly the same value as the integral over . ${\displaystyle f}$ ${\displaystyle \Omega }$ ${\displaystyle f}$ ${\displaystyle N}$ ${\displaystyle f}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega \setminus N}$

It must be noted that a zero set is only negligibly "small" in the sense of the measure. But it can also contain an infinite number of elements. For example, the set , i.e. the set of rational numbers as a subset of the real numbers, is a Lebesgue null set. The Dirichlet function ${\displaystyle \mathbb {Q} \subset \mathbb {R} }$

${\displaystyle f(x)={\begin{cases}1&x\in \mathbb {Q} \\0&{\text{sonst}}\end{cases}}}$

is therefore equal in the above sense to the function that constantly assumes the value zero ( zero function ), although there is no environment, however small, in which their values ​​match. A well-known uncountable (too equal) Lebesgue null set is the Cantor set . ${\displaystyle \mathbb {R} }$

Important properties of the Lebesgue integral

The integral is linear in ( space of integrable functions ), ie for integrable functions and and any can also be integrated and the following applies: ${\displaystyle {\mathcal {L}}^{1}}$ ${\displaystyle \ f\ }$ ${\displaystyle \ g\ }$ ${\displaystyle \alpha ,\beta \in \mathbb {R} }$ ${\displaystyle \ \alpha f+\beta g\ }$

${\displaystyle \int _{\Omega }(\alpha f+\beta g)\,\mathrm {d} \mu =\alpha \cdot \int _{\Omega }f\,\mathrm {d} \mu +\beta \cdot \int _{\Omega }g\,\mathrm {d} \mu }$

The integral is monotonic, ie if and are two measurable functions with , then applies ${\displaystyle \ f\ }$ ${\displaystyle \ g\ }$ ${\displaystyle f\leq g}$

${\displaystyle \int _{\Omega }f\ \mathrm {d} \mu \leq \int _{\Omega }g\ \mathrm {d} \mu }$.

The integral can be separated

${\displaystyle \int _{\Omega }f\ \mathrm {d} \mu =\int _{\Omega \backslash N}f\ \mathrm {d} \mu +\int _{N}f\ \mathrm {d} \mu }$

${\displaystyle \int _{\Omega \cup N}f\ \mathrm {d} \mu =\int _{\Omega }f\ \mathrm {d} \mu +\int _{N}f\ \mathrm {d} \mu -\int _{\Omega \cap N}f\ \mathrm {d} \mu }$

Is measurable with , then applies ${\displaystyle A\in {\mathcal {P}}(\Omega )}$ ${\displaystyle \ \mu (A)=0\ }$

${\displaystyle \ \int _{A}f\mathrm {d} \mu =0\ }$

Convergence theorems

One of the most important advantages of the Lebesgue integral is the very beautiful convergence theorems from a mathematical point of view. This concerns the interchangeability of limit value and integral in function sequences of the form . The most important convergence theorems are: ${\displaystyle \ (f_{n})_{n\in \mathbb {N} }\ }$

Theorem of monotonous convergence (Beppo Levi, 1906)

If there is a monotonically growing sequence of non-negative, measurable functions, then: ${\displaystyle f_{n}\colon (\Omega ,\Sigma ,\mu )\rightarrow ({\overline {\mathbb {R} }},B),n\in \mathbb {N} }$

${\displaystyle \int _{\Omega }\sup _{n\in \mathbb {N} }f_{n}\ \mathrm {d} \mu =\int _{\Omega }\lim _{n\rightarrow \infty }f_{n}\ \mathrm {d} \mu =\lim _{n\rightarrow \infty }\int _{\Omega }f_{n}\ \mathrm {d} \mu }$.

Theorem of the majorized (dominated) convergence (Henri Léon Lebesgue, 1910)

The sequence converges measurable functions -almost everywhere against the measurable function and the functions , , magnitude -almost everywhere by an integrable function is limited, then: ${\displaystyle f_{n}\colon (\Omega ,\Sigma ,\mu )\rightarrow ({\overline {\mathbb {R} }},B)}$ ${\displaystyle \mu }$ ${\displaystyle f\colon (\Omega ,\Sigma ,\mu )\rightarrow ({\overline {\mathbb {R} }},B)}$ ${\displaystyle f_{n}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \mu }$ ${\displaystyle g\colon (\Omega ,\Sigma ,\mu )\rightarrow ({\overline {\mathbb {R} }},B)}$

• ${\displaystyle f}$ can be integrated,
• ${\displaystyle \lim _{n\rightarrow \infty }\int _{\Omega }f_{n}\ \mathrm {d} \mu =\int _{\Omega }f\ \mathrm {d} \mu }$ and
• ${\displaystyle \lim _{n\rightarrow \infty }\int _{\Omega }|f-f_{n}|\ \mathrm {d} \mu =0.}$

Lemma of Fatou (Pierre Fatou, 1906)

Are , non-negative measurable functions, then: ${\displaystyle f_{n}\colon (\Omega ,\Sigma ,\mu )\rightarrow ({\overline {\mathbb {R} }},B)}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle \int _{\Omega }\liminf _{n\rightarrow \infty }f_{n}\ \mathrm {d} \mu \leq \liminf _{n\rightarrow \infty }\int _{\Omega }f_{n}\ \mathrm {d} \mu }$

Riemann and Lebesgue integral

Figure 2: Partial sums of the alternating harmonic series

In the case of the Lebesgue measure, the following applies: If a function is Riemann-integrable on a compact interval , it can also be Lebesgue-integrable and the values ​​of both integrals match. On the other hand, not every Lebesgue-integrable function can also be Riemann-integrable. ${\displaystyle \Omega =\mathbb {R} }$

However, an improperly Riemann integrable function does not have to be Lebesgue integrable as a whole ; however, the corresponding limit value of Lebesgue integrals exists according to the above remarks and provides the same value as for the Riemann integrals. But if Riemann can not be properly integrated, then Lebesgue can even be integrated as a whole. ${\displaystyle |f|}$ ${\displaystyle f}$

One can easily give an example of an improperly Riemann integrable function that is not Lebesgue integrable: if a step function with the areas 1 , -1/2 , 1/3 , etc., is improperly Riemann integrable. Because the integral corresponds to the alternating harmonic series . If Lebesgue were integrable, then it would apply. However, this is not the case because the harmonic series is divergent. Hence the corresponding Lebesgue integral does not exist . The situation is shown in Figure 2. ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle \textstyle \int _{\mathbb {R} ^{+}}|f|\,\mathrm {d} \lambda <\infty }$

The reverse case of a Lebesgue integrable function that is not Riemann integrable is more important.

The best-known example of this is the Dirichlet function :

${\displaystyle f\colon [0,1]\rightarrow [0,1]}$

${\displaystyle x\mapsto {\begin{cases}1&,x\in \mathbb {Q} \\0&,{\text{sonst}}.\end{cases}}}$

${\displaystyle f}$is not Riemann-integrable, since all lower sums are always 0 and all upper sums are always 1. But since the set of rational numbers in the set of real numbers is a Lebesgue null set, the function is 0 almost everywhere . So the Lebesgue integral exists and has the value 0. ${\displaystyle \mathbb {Q} \,,}$

The main difference in the procedure for the integration according to Riemann or Lebesgue is that with the Riemann integral the domain of definition ( abscissa ), with the Lebesgue integral the image set ( ordinate ) of the function is divided. The above examples already show that this difference can turn out to be decisive.

Henri Lebesgue said of the comparison between the Riemann and Lebesgue integral:

“You can say that with Riemann's approach you behave like a merchant without a system, who counts coins and banknotes in the order in which he gets them in his hand; while we act like a prudent businessman who says:

I have coins to a crown, makes , ${\displaystyle m(E_{1})}$ ${\displaystyle 1\cdot m(E_{1})}$

I have two kroner coins , ${\displaystyle m(E_{2})}$ ${\displaystyle 2\cdot m(E_{2})}$

I have five kroner coins , ${\displaystyle m(E_{3})}$ ${\displaystyle 5\cdot m(E_{3})}$

etc., so I have a total . The two methods certainly lead the merchant to the same result because - however rich he is - he only has to count a finite number of banknotes; but for us, who have an infinite number of indivisibles to add, the difference between the two approaches is essential. " ${\displaystyle S=1\cdot m(E_{1})+2\cdot m(E_{2})+5\cdot m(E_{3})+\ldots }$

- Henri Lebesgue, 1926 : after Jürgen Elstrodt

Bochner integral

→ Main article : Bochner integral

The Bochner integral is a direct generalization of the Lebesgue integral for Banach space- valued functions. It has almost all the properties of the Lebesgue integral, such as the theorem of majorized convergence.

literature

• Jürgen Elstrodt : Measure and integration theory . 2nd, corrected edition. Springer, Heidelberg 1999, ISBN 3-540-65420-8 , IV. The Lebesgue integral, p. 118-160 .
• Walter Rudin : Analysis . 2nd, corrected edition. Oldenbourg, Munich / Vienna 2002, ISBN 3-486-25810-9 , 11. The Lebesgue theory, p. 353-392 .
• Klaus D. Schmidt: Measure and Probability . Springer-Verlag, Berlin / Heidelberg 2009, ISBN 978-3-540-89729-3 , 8th Lebesgue integral, 9th calculation of the Lebesgue integral, p. 109-190 .