- Lebesgue integration
In

mathematics , the "integral " of a non-negative function can be regarded in the simplest case as thearea between the graph of that function and the "x"-axis.**Lebesgue integration**is a mathematical construction that extends theintegral to a larger class of functions; it also extends the domains on which these functions can be defined. It had long been understood that for non-negative functions with a smooth enough graph (such as continuous functions on closed bounded intervals) the "area under the curve" could be defined as the integral and computed using techniques of approximation of the region bypolygon s. However, as the need to consider more irregular functions arose (for example, as a result of the limiting processes ofmathematical analysis and the mathematicaltheory of probability ) it became clear that more careful approximation techniques would be needed in order to define a suitable integral.The Lebesgue integral plays an important role in the branch of mathematics called

real analysis and in many other fields in the mathematical sciences.The Lebesgue integral is named for

Henri Lebesgue (1875 -1941 ) who introduced the integral in harv|Lebesgue|1904. His last name is pronounced|ləˈbɛg, approximately "luh beg".The term "Lebesgue integration" may refer either to the general theory of integration of a function with respect to a general measure, as introduced by Lebesgue, or to the specific case of integration of a function defined on a sub-domain of the

real line with respect toLebesgue measure .**Introduction**The integral of a function "f" between limits "a" and "b" can be interpreted as the area 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

rigour in mathematics in thenineteenth century , attempts were made to put the integral calculus on a firm foundation. TheRiemann integral , proposed byBernhard 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 areas 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, Riemann integration does not interact well with taking limits of sequences of functions, making such limiting processes difficult to analyze. This is of prime importance, for instance, in the study of

Fourier series ,Fourier transform s 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 areas 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, theDirichlet function , which is 0 where its argument is irrational and 1 otherwise, has a Lebesgue integral, but it does not have a Riemann integral.**Construction of the Lebesgue integral**The discussion that follows parallels the most common expository approach to the Lebesgue integral. In this approach, the theory of integration has two distinct parts:

# A theory of measurable sets and measures on these sets.

# A theory of measurable functions and integrals on these functions.**Measure theory**Measure theory initially was created to provide a detailed analysis 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 was shown by later developments inset theory (seenon-measurable set ), it is actually impossible to assign a length to all subsets of**R**in a way which preserves some natural additivity and translation invariance properties. This suggests that picking out a suitable class of "measurable" subsets is an essential prerequisite.Of course, the Riemann integral uses the notion of length implicitly. 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 certain subsets "X" of a set "E" which satisfies a certain list of properties. These properties can be shown to hold in many different cases.

The theory of measurable sets and measure (including definition and construction of such measures) is discussed in other articles. See measure.

**Integration**As usual we start with a

measure space , ("E","X",μ). In this, "E" is just a set, "X" is a σ-algebra of subsets of "E" and μ is a (non-negative) measure on "X" of subsets of "E".For example, "E" can be Euclidean "n"-space

**R**^{"n"}or some Lebesgue measurable subset of it, "X" will be the σ-algebra of all Lebesgue measurable subsets of "E", and μ will be theLebesgue measure . In the mathematical theory of probability μ will be aprobability measure on a probability space "E".In Lebesgue's theory, integrals are limited to a class of functions called

measurable function s. A function "f" is measurable if the pre-image of every closed interval is in "X"::$f^\{-1\}(\; [a,b]\; )\; in\; X\; mbox\{\; for\; all\; \}amath>$

It can be shown that this is equivalent to requiring that the pre-image of any Borel subset of

**R**be in "X". We will make this assumption from now on. The set of measurable functions is closed under algebraic operations, but more importantly the class is closed under various kinds of pointwise sequential limits:: $liminf\_\{k\; in\; mathbb\{N\; f\_k,\; quad\; limsup\_\{k\; in\; mathbb\{N\; f\_k$

are measurable if the original sequence {"f"

_{"k"}}, where "k" $in$**N**, consists of measurable functions.We build up an integral

: $int\_E\; f\; d\; mu\; =\; int\_E\; fleft(x\; ight)muleft(dx\; ight)$

for measurable real-valued functions "f" defined on "E" in stages:

**Indicator functions**: To assign a value to the integral of theindicator function of a measurable set "S" consistent with the given measure μ, the only reasonable choice is to set::$int\; 1\_S\; d\; mu\; =\; mu\; (S)$

**Simple functions**: We extend by linearity to thelinear span of indicator functions::$int\; igg(sum\_k\; a\_k\; 1\_\{S\_k\}igg)\; d\; mu\; =\; sum\_k\; a\_k\; int\; 1\_\{S\_k\}d\; mu\; =\; sum\_k\; a\_k\; ,\; mu(S\_k)$

where the sum is finite and the coefficients "a"

_{"k"}are real numbers. Such a finitelinear combination of indicator functions is called a "simple function ". Even if a simple function can be written in many ways as a linear combination of indicator functions, the integral will always be the same.If "E" is a measurable set and "s" a measurable simple function one defines:$int\_E\; s\; ,\; dmu\; =\; sum\_k\; a\_k\; ,\; mu(S\_k\; cap\; E)$

**Non-negative functions**: Let "f" be a non-negative measurable function on "E" which we allow to attain the value +∞, in other words, "f" takes non-negative values in theextended real number line . We define:$int\_E\; f,dmu\; =\; supleft\{,int\_E\; s,dmu\; :\; sle\; f,\; s\; mbox\{simple\},\; ight\}.$

We need to show this integral coincides with the preceding one, defined on the set of simple functions. There is also the question of whether this corresponds in any way to a Riemann notion of integration. It is possible to prove that the answer to both questions is yes.

We have defined the integral of "f" for any non-negative extended real-valued measurable function on "E". For some functions ∫"f" will be infinite.

**Signed functions**: 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:$f\; =\; f^+\; -\; f^-,\; quad$

where

:$f^+(x)\; =\; left\{egin\{matrix\}\; f(x)\; mbox\{if\}\; quad\; f(x)\; 0\; \backslash \; 0\; mbox\{otherwise\}\; end\{matrix\}\; ight.$

:$f^-(x)\; =\; left\{egin\{matrix\}\; -f(x)\; mbox\{if\}\; quad\; f(x)\; 0\; \backslash \; 0\; mbox\{otherwise\}\; end\{matrix\}\; ight.$

Note that both "f"

^{+}and "f"^{−}are non-negative functions. Also note that:$|f|\; =\; f^+\; +\; f^-.\; quad$

If

:$int\; |f|\; d\; mu\; <\; infty,$

then "f" is called "Lebesgue integrable". In this case, both integrals satisfy

:$int\; f^+\; d\; mu\; <\; infty,\; quad\; int\; f^-\; d\; mu\; <\; infty,$

and it makes sense to define

:$int\; f\; d\; mu\; =\; int\; f^+\; d\; mu\; -\; int\; f^-\; d\; mu$

It turns out that this definition gives the desirable properties of the integral.

**Complex valued functions**can be similarly integrated, by considering the real part and the imaginary part separately.**Intuitive interpretation**To get some intuition about the different approaches to integration, let us imagine that it is desired to find a mountain's volume (above sea level).

**The Riemann-Darboux approach**: Divide the base of the mountain into a grid of 1 meter squares (acadaster , in the language of land surveyors). Measure the altitude of the mountain at the center of each square. The volume on a single grid square is approximately 1x1x(altitude), so the total volume is the sum of the altitudes.**The Lebesgue approach**: Draw acontour map of the mountain, where each contour is 1 meter of altitude apart. The volume of earth contained in a single contour is approximately that contour's area times its thickness. So the total volume is the sum of the areas of the contours.Folland [

*Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 1984, p. 56.*] 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".See also Properties of simple functions.

**Example**Consider the

indicator function of the rational numbers, 1_{Q}. This function isnowhere continuous .*

**$1\_\{mathbb\; Q\}$ is not Riemann-integrable on**[0,1] : No matter how the set [0,1] is partitioned into subintervals, each partition will contain at least one rational and at least one irrational number, since rationals and irrationals are both dense in the reals. Thus the upperDarboux sum s will all be one, and the lower Darboux sums will all be zero.*

**$1\_\{mathbb\; Q\}$ is Lebesgue-integrable on**[0,1] using theLebesgue measure : Indeed it is the indicator function of the rationals so by definition::$int\_\{\; [0,1]\; \}\; 1\_\{mathbb\{Q\; ,\; d\; mu\; =\; mu(mathbb\{Q\}\; cap\; [0,1]\; )\; =\; 0,$

:since $mathbb\; Q$ is

countable .**Limitations of the Riemann integral**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 , many analytical problems involving integrals came up whose satisfactory solution required exchanging infinite summations of functions and integral signs. However, the conditions under which the integrals: $sum\_k\; int\; f\_k(x)\; dx$ and $int\; igg\; [sum\_k\; f\_k(x)\; igg]\; dx$

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.

**Failure of monotone convergence**. As shown above, the indicator function 1_{Q}on the rationals is not Riemann integrable. In particular, theMonotone convergence theorem fails. To see why, let {"a"_{"k"}} be an enumeration of all the rational numbers in [0,1] (they arecountable so this can be done.) Then let :$g\_k(x)\; =\; left\{egin\{matrix\}\; 1\; mbox\{if\; \}\; x\; =\; a\_j,\; jleq\; k\; \backslash 0\; mbox\{otherwise\}\; end\{matrix\}\; ight.$The function "g"

_{"k"}is zero everywhere except on a finite set of points, hence its Riemann integral is zero. The sequence "g"_{"k"}is also clearly non-negative and monotonically increasing to 1_{Q}, which is not Riemann integrable.**Unsuitability for unbounded intervals**. The Riemann integral can only integrate functions on a bounded interval. It can however be extended to unbounded intervals by taking limits, so long as this doesn't yield an answer such as $infty\; -infty$.**Basic theorems of the Lebesgue integral**The Lebesgue integral does not distinguish between functions which only differ on a set of μ-measure zero. To make this precise, functions "f", "g" are said to be equal

almost everywhere (or equal a.e.)if and only if :$mu(\{x\; in\; E:\; f(x)\; eq\; g(x)\})\; =\; 0$

* If "f", "g" are non-negative functions (possibly assuming the value +∞) such that "f" = "g" almost everywhere, then

:$int\; f\; d\; mu\; =\; int\; g\; d\; mu.$

* If "f", "g" are functions such that "f" = "g" almost everywhere, then "f" is Lebesgue integrable if and only if "g" is Lebesgue integrable and the integrals of "f" and "g" are the same.

The Lebesgue integral has the following properties:

Linearity: If "f" and "g" are Lebesgue integrable functions and "a" and "b" are real numbers, then "af" + "bg" is Lebesgue integrable and

:$int\; (a\; f\; +\; bg)\; d\; mu\; =\; a\; int\; f\; dmu\; +\; b\; int\; g\; dmu$

Monotonic ity: If "f" ≤ "g", then:$int\; f\; d\; mu\; leq\; int\; g\; d\; mu.$

Monotone convergence theorem: Suppose {"f"

_{"k"}}_{"k" $in$ N}is a sequence of real, non-negative measurable functions such that:$f\_k(x)\; leq\; f\_\{k+1\}(x)\; quad\; forall\; kin\; mathbb\{N\},\; forall\; x\; in\; E.$

Then

:$lim\_k\; int\; f\_k\; d\; mu\; =\; int\; sup\_k\; f\_k\; d\; mu.$

Note: The value of any of the integrals is allowed to be infinite.

Fatou's lemma : If {"f"_{"k"}}_{"k" $in$ N}is a sequence of real, non-negative measurable functions, then:$int\; liminf\_k\; f\_k\; d\; mu\; leq\; liminf\_k\; int\; f\_k\; d\; mu.$

Again, the value of any of the integrals may be infinite.

Dominated convergence theorem : If {"f"_{"k"}}_{"k" $in$ N}is a sequence of complex measurable functions with pointwise limit "f", and if there is a Lebesgue integrable function "g" (i.e, "g $in$ L"^{1}) such that |"f"_{"k"}| ≤ "g" for all "k", then "f" is Lebesgue integrable and:$lim\_k\; int\; f\_k\; d\; mu\; =\; int\; f\; d\; mu.$

**Proof techniques**To illustrate some of the proof techniques used in Lebesgue integration theory, we sketch a proof of the above mentioned Lebesgue monotone convergence theorem:

Let {"f"

_{"k"}}_{"k" $in$ N}be a non-decreasing sequence of non-negative measurable functions and put:$f\; =\; sup\_\{k\; in\; mathbb\{N\; f\_k$

By the monotonicity property of the integral, it is immediate that:

: $int\; f\; d\; mu\; geq\; lim\_k\; int\; f\_k\; d\; mu$

and the limit on the right exists, since the sequence is monotonic.

We now prove the inequality in the other direction (which also follows from Fatou's lemma), that is

:$int\; f\; d\; mu\; leq\; lim\_k\; int\; f\_k\; d\; mu.$

It follows from the definition of integral, that there is a non-decreasing sequence "g"

_{"n"}of non-negative simple functions which converges to "f" pointwise almost everywhere and such that:$lim\_k\; int\; g\_k\; d\; mu\; =\; int\; f\; d\; mu.$

Therefore, it suffices to prove that for each "k" $in$

**N**,:$int\; g\_k\; d\; mu\; leq\; lim\_j\; int\; f\_j\; d\; mu.$

We will show that if "g" is a simple function and

:$lim\_j\; f\_j(x)\; geq\; g(x)$

almost everywhere, then

:$lim\_j\; int\; f\_j\; d\; mu\; geq\; int\; g\; d\; mu.$

By breaking up the function "g" into its constant value parts, this reduces to the case in which "g" is the indicator function of a set. The result we have to prove is then

:Suppose "A" is a measurable set and {"f"

_{"k"}}_{"k" $in$ N}is a nondecreasing sequence of measurable functions on "E" such that::$lim\_n\; f\_n\; (x)\; geq\; 1$

:for almost all "x" $in$ "A". Then

::$lim\_n\; int\; f\_n\; dmu\; geq\; mu(A).$

To prove this result, fix ε > 0 and define the sequence ofmeasurable sets

:$B\_n\; =\; \{x\; in\; A:\; f\_n(x)\; geq\; 1\; -\; epsilon\; \}.$

By monotonicity of the integral, it follows that for any"n" ∈

**N**,:$mu(B\_n)\; (1\; -\; epsilon)\; =\; int\; (1\; -\; epsilon)1\_\{B\_n\}\; d\; mu\; leq\; int\; f\_n\; d\; mu$

Because of the fact that almost every "x" will be in $B\_n$ for large enough "n", we have

:$igcup\_i\; B\_i\; =\; A,$

up to a set of measure 0. Thus by countable additivity of μ

:$mu(A)\; =\; lim\_n\; mu(B\_n)\; leq\; lim\_n\; (1\; -\; epsilon)^\{-1\}\; int\; f\_n\; dmu.$

As this is true for any positive ε the result follows.

**Alternative formulations**It is possible to develop the integral with respect to the Lebesgue measure without relying on the full machinery of measure theory. One such approach is provided by

Daniell integral .There is also an alternative approach to developing the theory of integration via methods of

functional analysis . The Riemann integral exists for any continuous function "f" of compact support defined on**R**^{n}(or a fixed open subset). Integrals of more general functions can be built starting from these integrals. Let "C_{c}" be the space of all real-valued compactly supported continuous functions of**R**. Define a norm on "C_{c}" by: $|f|\; =\; int\; |f(x)|\; dx$

Then "C

_{c}" is a normed vector space (and in particular, it is a metric space.) All metric spaces have Hausdorff completions, so let "L"^{1}be its completion. This space is isomorphic to the space of Lebesgue integrable functions modulo the subspace of functions with integral zero. Furthermore, the Riemann integral ∫ is auniformly continuous functional with respect to the norm on "C_{c}", which is dense in "L"^{1}. Hence ∫ has a unique extension to all of "L"^{1}. This integral is precisely the Lebesgue integral.This approach can be generalised to build the theory of integration with respect to

Radon measure s onlocally compact space s. It is the approach adopted byBourbaki (2004); for more details see Radon measures on locally compact spaces.**Applications, e.g. in functional analysis**Finally one should of course mention that many statements on topological vector spaces(e.g. Hilbert or

Banach spaces ) and on limiting procedures therein (e.g. strong or weak convergence) are essentially simplified by using from thebeginning the Lebesgue integral.**See also***

null set

* integration

* measure

*sigma-algebra

*Lebesgue space

*Lebesgue-Stieltjes integration

*Henstock-Kurzweil integral **Notes****References*** cite book

last = Bartle

first = Robert G.

title = The elements of integration and Lebesgue measure

series = Wiley Classics Library

publisher = John Wiley & Sons Inc.

location = New York

year = 1995

pages = pp. xii+179

isbn = 0-471-04222-6 MathSciNet|id=1312157* cite book

last = Bourbaki

first = Nicolas

authorlink = Nicolas Bourbaki

title = Integration. I. Chapters 1–6. Translated from the 1959, 1965 and 1967 French originals by Sterling K. Berberian

series = Elements of Mathematics (Berlin)

publisher= Springer-Verlag

location = Berlin

year = 2004

pages = pp. xvi+472

isbn = 3-540-41129-1 MathSciNet|id=2018901* cite book

last = Dudley

first = Richard M.

title = Real analysis and probability

series = The Wadsworth &ammp; Brooks/Cole Mathematics Series

publisher = Wadsworth & Brooks/Cole Advanced Books & Software

location = Pacific Grove, CA

year = 1989

pages = pp. xii+436

isbn = 0-534-10050-3 MathSciNet|id=982264 Very thorough treatment, particularly for probabilists with good notes and historical references.* cite book

last = Folland

first = Gerald B.

title = Real analysis: Modern techniques and their applications

series = Pure and Applied Mathematics (New York)

edition = Second edition

publisher = John Wiley & Sons Inc.

location = New York

year = 1999

pages = pp. xvi+386

isbn = 0-471-31716-0 MathSciNet|id=1681462* cite book

last = Halmos

first = Paul R.

authorlink = Paul Halmos

title = Measure Theory

publisher = D. Van Nostrand Company, Inc.

location = New York, N. Y.

year = 1950

pages = pp. xi+304 MathSciNet|id=0033869 A classic, though somewhat dated presentation.* citation

last = Lebesgue

first = Henri

authorlink = Henri Lebesgue

title = Leçons sur l'intégration et la recherche des fonctions primitives

publisher = Gauthier-Villars

year = 1904

publication-place = Paris* cite book

last = Lebesgue

first = Henri

authorlink = Henri Lebesgue

title = Oeuvres scientifiques (en cinq volumes)

publisher = Institut de Mathématiques de l'Université de Genève

location = Geneva

year = 1972

pages = pp. 405

language = French MathSciNet|id=0389523* cite book

last = Loomis

first = Lynn H.

title = An introduction to abstract harmonic analysis

publisher = D. Van Nostrand Company, Inc.

location = Toronto-New York-London

year = 1953

pages = pp. x+190 MathSciNet|id=0054173 Includes a presentation of the Daniell integral.* cite book

last = Munroe

first = M. E.

title = Introduction to measure and integration

publisher = Addison-Wesley Publishing Company Inc.

location = Cambridge, Mass.

year = 1953

pages = pp. x+310 MathSciNet|id=0053186 Good treatment of the theory of outer measures.* cite book

last = Royden

first = H. L.

title = Real analysis

edition = Third edition

publisher = Macmillan Publishing Company

location = New York

year = 1988

pages = pp. xx+444

isbn = 0-02-404151-3 MathSciNet|id=1013117* cite book

last = Rudin

first = Walter

authorlink = Walter Rudin

title = Principles of mathematical analysis

edition = Third edition

series = International Series in Pure and Applied Mathematics

publisher = McGraw-Hill Book Co.

location = New York

year = 1976

pages = pp. x+342 MathSciNet|id=0385023 Known as "Little Rudin", contains the basics of the Lebesgue theory, but does not treat material such asFubini's theorem .* cite book

last = Rudin

first = Walter

title = Real and complex analysis

publisher = McGraw-Hill Book Co.

location = New York

year = 1966

pages = pp. xi+412 MathSciNet|id=0210528 Known as "Big Rudin". A complete and careful presentation of the theory. Good presentation of the Riesz extension theorems. However, there is a minor flaw (in the first edition) in the proof of one of the extension theorems, the discovery of which constitutes exercise 21 of Chapter 2.* cite book

last = Shilov

first = G. E.

coauthors = Gurevich, B. L.

title = Integral, measure and derivative: a unified approach. Translated from the Russian and edited by Richard A. Silverman

series = Dover Books on Advanced Mathematics

publisher = Dover Publications Inc.

loaction = New York

year = 1977

pages = pp. xiv+233

isbn = 0-486-63519-8 MathSciNet|id=0466463 Emphasizes theDaniell integral .

*Wikimedia Foundation.
2010.*

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