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Lebesgue integral

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(Redirected fromLebesgue integration)
Method of integration
The integral of a positive function can be interpreted as the area under a curve.
Part of a series of articles about
Calculus
∫abf′(t)dt=f(b)−f(a){\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}

Inmathematics, theintegral of a non-negativefunction of a single variable can be regarded, in the simplest case, as thearea between thegraph of that function and theX axis. TheLebesgue integral, named afterFrench mathematicianHenri Lebesgue, is one way to make this concept rigorous and to extend it to more general functions.

The Lebesgue integral is more general than theRiemann integral, which it largely replaced inmathematical 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 inprobability theory.

The termLebesgue integration can mean either the general theory of integration of a function with respect to a generalmeasure, as introduced by Lebesgue, or the specific case of integration of a function defined on a sub-domain of thereal line with respect to theLebesgue measure.

Introduction

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The integral of a positive real functionf between boundariesa andb can be interpreted as the area under the graph off, betweena andb. This notion of area fits some functions, mainlypiecewise continuous functions, includingelementary functions, for examplepolynomials. However, the graphs of other functions, for example theDirichlet 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 towardrigor in mathematics in the nineteenth century, mathematicians attempted to put integral calculus on a firm foundation. TheRiemann integral—proposed byBernhard 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 ofFourier 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 themonotone convergence theorem anddominated 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 is1 where its argument isrational and0 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 theunit interval, the probability of picking a rational number should be zero.

Lebesgue summarized his approach to integration in a letter toPaul Montel:

I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.

— Source: (Siegmund-Schultze 2008)

The insight is that one should be able to rearrange the values of a function freely, while preserving the value of the integral. This process of rearrangement can convert a verypathological function into one that is "nice" from the point of view of integration, and thus let such pathological functions be integrated.

Intuitive interpretation

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A measurable function is shown, together with the set{x :f(x) >t} (on thex-axis). The Lebesgue integral is obtained by slicing along they-axis, using the 1-dimensional Lebesgue measure to measure the "width" of the slices.

Folland (1999) summarizes the difference between the Riemann and Lebesgue approaches thus: "to compute the Riemann integral off, one partitions the domain[a,b] into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range off".

For the Riemann integral, thedomain 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 formf(x)dx wheref(x) is the height of a rectangle anddx is its width.

For the Lebesgue integral, therange 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 off, of heightdy, is equal to the measure of the slab's width timesdy:μ({x∣f(x)>y})dy.{\displaystyle \mu \left(\{x\mid f(x)>y\}\right)\,dy.}The Lebesgue integral may then bedefined 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.

Simple functions

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Riemannian (top) vs. Lebesgue (bottom) integration of smoothed COVID-19 daily case data from Serbia (Summer-Fall 2021)

An equivalent way to introduce the Lebesgue integral is to use so-calledsimple functions, which generalize the step functions of Riemann integration. Consider, for example, determining the cumulative COVID-19 case count from a graph of smoothed cases each day (right).

The Riemann–Darboux approach
Partition the domain (time period) into intervals (eight, in the example) and construct bars with heights that meet the graph. The cumulative count is found by summing, over all bars, the product of interval width (time in days) and the bar height (cases per day).
The Lebesgue approach
Choose a finite number of target values (eight, in the example) in the range of the function. By constructing bars with heights equal to these values, but below the function, they imply a partitioning of the domain into the same number of subsets (subsets, indicated by color in the example, need not be connected). This is a "simple function", as described below. The cumulative count is found by summing, over all subsets of the domain, the product of themeasure on that subset (total time in days) and the bar height (cases per day).

Relation between the viewpoints

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One can think of the Lebesgue integral either in terms ofslabs orsimple 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.

Theslabs viewpoint makes it easy to define the Lebesgue integral, in terms of basic calculus. Suppose thatf is a (Lebesgue measurable) function, taking non-negative values (possibly including+∞). Define the distribution function off as the "width of a slab", i.e.,F(y)=μ{x|f(x)>y}.{\displaystyle F(y)=\mu \{x|f(x)>y\}.}ThenF(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 bedefined by∫fdμ=∫0∞F(y)dy{\displaystyle \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+∞ ifF(y) = +∞ on a neighborhood of0).

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

Measure theory

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Further information:Measure (mathematics)

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 ofR have a length. As laterset theory developments showed (seenon-measurable set), it is actually impossible to assign a length to all subsets ofR in a way that preserves some natural additivity and translation invariance properties. This suggests that picking out a suitable class ofmeasurable 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 quantityb −a is the length of the base of the rectangle andd −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 isaxiomatic. This means that a measure is any functionμ defined on a certain classX of subsets of a setE, which satisfies a certain list of properties. These properties can be shown to hold in many different cases.

Measurable functions

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We start with ameasure space(E,X,μ) whereE is aset,X is aσ-algebra of subsets ofE, andμ is a (non-negative)measure onE defined on the sets ofX.

For example,E can beEuclideann-spaceRn or someLebesgue measurable subset of it,X is theσ-algebra of all Lebesgue measurable subsets ofE, andμ is the Lebesgue measure. In the mathematical theory of probability, we confine our study to aprobability measure μ, which satisfiesμ(E) = 1.

Lebesgue's theory defines integrals for a class of functions calledmeasurable functions. A real-valued functionf onE is measurable if thepre-image of every interval of the form(t, ∞) is inX:{x∣f(x)>t}∈X∀t∈R.{\displaystyle \{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 anyBorel subset ofR be inX. The set of measurable functions is closed under algebraic operations, but more importantly it is closed under various kinds ofpoint-wise sequential limits:supk∈Nfk,lim infk∈Nfk,lim supk∈Nfk{\displaystyle \sup _{k\in \mathbb {N} }f_{k},\quad \liminf _{k\in \mathbb {N} }f_{k},\quad \limsup _{k\in \mathbb {N} }f_{k}}are measurable if the original sequence(fk), wherek ∈N, consists of measurable functions.

There are several approaches for defining an integral for measurable real-valued functionsf defined onE, and several notations are used to denote such an integral.∫Efdμ=∫Ef(x)dμ(x)=∫Ef(x)μ(dx).{\displaystyle \int _{E}f\,d\mu =\int _{E}f(x)\,d\mu (x)=\int _{E}f(x)\,\mu (dx).}

Following the identification inDistribution theory of measures with distributions of order0, or withRadon measures, one can also use adual pair notation and write the integral with respect toμ in the form⟨μ,f⟩.{\displaystyle \langle \mu ,f\rangle .}

Definition

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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.

Via simple functions

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Approximating a function by a simple function.

One approach to constructing the Lebesgue integral is to make use of so-calledsimple functions: finite, real linear combinations ofindicator functions. Simple functions that lie directly underneath a given functionf can be constructed by partitioning the range off into a finite number of layers. The intersection of the graph off with a layer identifies a set of intervals in the domain off, 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 off 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 appropriatesupremum 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.[1]

Indicator functions

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To assign a value to the integral of theindicator function1S of a measurable setS consistent with the givenmeasureμ, the only reasonable choice is to set:∫1Sdμ=μ(S).{\displaystyle \int 1_{S}\,d\mu =\mu (S).}

Notice that the result may be equal to+∞, unlessμ is afinite measure.

Simple functions

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A finitelinear combination of indicator functions∑kak1Sk{\displaystyle \sum _{k}a_{k}1_{S_{k}}}where the coefficientsak are real numbers andSk are disjoint measurable sets, is called a measurablesimple function. We extend the integral by linearity tonon-negative measurable simple functions. When the coefficientsak are positive, we set∫(∑kak1Sk)dμ=∑kak∫1Skdμ=∑kakμ(Sk){\displaystyle \int \left(\sum _{k}a_{k}1_{S_{k}}\right)\,d\mu =\sum _{k}a_{k}\int 1_{S_{k}}\,d\mu =\sum _{k}a_{k}\,\mu (S_{k})}whether this sum is finite or+∞. A simple function can be written in different ways as a linear combination of indicator functions, but the integral will be the same by the additivity of measures.

Some care is needed when defining the integral of areal-valued simple function, to avoid the undefined expression∞ − ∞: one assumes that the representationf=∑kak1Sk{\displaystyle f=\sum _{k}a_{k}1_{S_{k}}}is such thatμ(Sk) < ∞ wheneverak ≠ 0. Then the above formula for the integral off makes sense, and the result does not depend upon the particular representation off satisfying the assumptions.[2] (It is important that the representation be afinite linear combination, i.e. thatk only take on a finite number of values.)

IfB is a measurable subset ofE ands is a measurable simple function one defines∫Bsdμ=∫1Bsdμ=∑kakμ(Sk∩B).{\displaystyle \int _{B}s\,\mathrm {d} \mu =\int 1_{B}\,s\,\mathrm {d} \mu =\sum _{k}a_{k}\,\mu (S_{k}\cap B).}

Non-negative functions

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Letf be a non-negative measurable function onE, which we allow to attain the value+∞, in other words,f takes non-negative values in theextended real number line. We define∫Efdμ=sup{∫Esdμ:0≤s≤f, s simple}.{\displaystyle \int _{E}f\,d\mu =\sup \left\{\,\int _{E}s\,d\mu :0\leq s\leq f,\ s\ {\text{simple}}\,\right\}.}

We need to show this integral coincides with the preceding one, defined on the set of simple functions, whenE is a segment[a,b]. 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 off for any non-negative extended real-valued measurable function onE. For some functions, this integral∫Efdμ{\textstyle \int _{E}f\,d\mu } is infinite.

It is often useful to have a particular sequence of simple functions that approximates the Lebesgue integral well (analogously to a Riemann sum). For a non-negative measurable functionf, letsn(x) be the simple function whose value isk/2n wheneverk/2n ≤f(x) < (k + 1)/2n, fork a non-negative integer less than, say,4n. Then it can be proven directly that∫fdμ=limn→∞∫sndμ{\displaystyle \int f\,d\mu =\lim _{n\to \infty }\int s_{n}\,d\mu }and that the limit on the right hand side exists as an extended real number. This bridges the connection between the approach to the Lebesgue integral using simple functions, and the motivation for the Lebesgue integral using a partition of the range.

Signed functions

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To handle signed functions, we need a few more definitions. Iff is a measurable function of the setE to the reals (including±∞), then we can writef=f+−f−,{\displaystyle f=f^{+}-f^{-},}wheref+(x)={f(x)−if f(x)>0,0otherwise,f−(x)={−f(x)if f(x)<0,0otherwise.{\displaystyle {\begin{aligned}f^{+}(x)&={\begin{cases}f(x){\hphantom {-}}&{\text{if }}f(x)>0,\\0&{\text{otherwise}},\end{cases}}\\f^{-}(x)&={\begin{cases}-f(x)&{\text{if }}f(x)<0,\\0&{\text{otherwise}}.\end{cases}}\end{aligned}}}

Note that bothf+ andf− are non-negative measurable functions. Also note that|f|=f++f−.{\displaystyle |f|=f^{+}+f^{-}.}

We say that the Lebesgue integral of the measurable functionfexists, oris defined if at least one of∫f+dμ{\textstyle \int f^{+}\,d\mu } and∫f−dμ{\textstyle \int f^{-}\,d\mu } is finite:min(∫f+dμ,∫f−dμ)<∞.{\displaystyle \min \left(\int f^{+}\,d\mu ,\int f^{-}\,d\mu \right)<\infty .}

In this case wedefine∫fdμ=∫f+dμ−∫f−dμ.{\displaystyle \int f\,d\mu =\int f^{+}\,d\mu -\int f^{-}\,d\mu .}

If∫|f|dμ<∞,{\displaystyle \int |f|\,\mathrm {d} \mu <\infty ,}we say thatf isLebesgue integrable. That is,f belongs to thespaceL1.[3]

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

Via improper Riemann integral

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See also:Layer cake representation

Assuming thatf is measurable and non-negative, the functionf∗(t) =def μ({x∈E∣f(x)>t}).{\displaystyle 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 theimproper Riemann integral off∗:[4]∫Efdμ =def ∫0∞f∗(t)dt.{\displaystyle \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.[5][6]

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

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Complex-valued functions can be similarly integrated, by considering the real part and the imaginary part separately.[7]

Ifh =f +ig for real-valued integrable functionsf,g, then the integral ofh is defined by∫hdμ=∫fdμ+i∫gdμ.{\displaystyle \int h\,d\mu =\int f\,d\mu +i\int g\,d\mu .}

The function is Lebesgue integrable if and only if itsabsolute value is Lebesgue integrable (seeAbsolutely integrable function).

Example

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Consider theindicator function of the rational numbers,1Q, also known as the Dirichlet function. This function isnowhere continuous.

Domain of integration

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A technical issue in Lebesgue integration is that the domain of integration is defined as aset (a subset of a measure space), with no notion of orientation. In elementary calculus, one defines integration with respect to anorientation:∫baf=−∫abf.{\displaystyle \int _{b}^{a}f=-\int _{a}^{b}f.}Generalizing this to higher dimensions yields integration ofdifferential forms. By contrast, Lebesgue integration provides an alternative generalization, integrating over subsets with respect to a measure; this can be notated as∫Afdμ=∫[a,b]fdμ{\displaystyle \int _{A}f\,d\mu =\int _{[a,b]}f\,d\mu }to indicate integration over a subsetA. For details on the relation between these generalizations, seeDifferential form § Relation with measures. The main theory linking these ideas is that ofhomological integration (sometimes called geometric integration theory), pioneered byGeorges de Rham andHassler Whitney.[8]

Limitations of the Riemann integral

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With the advent ofFourier series, many analytical problems involving integrals arose whose satisfactory solution required interchanging limit processes and integral signs. However, the conditions under which the integrals∑k∫fk(x)dxand∫[∑kfk(x)]dx{\displaystyle \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.

Failure of monotone convergence

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As shown above, theindicator function1Q on the rationals is not Riemann integrable. In particular, theMonotone convergence theorem fails. To see why, let{ak} be an enumeration of all the rational numbers in[0, 1] (they arecountable so this can be done). Then letgk(x)={1if x=aj,j≤k0otherwise{\displaystyle g_{k}(x)={\begin{cases}1&{\text{if }}x=a_{j},j\leq k\\0&{\text{otherwise}}\end{cases}}}

The functiongk is zero everywhere, except on a finite set of points. Hence its Riemann integral is zero. Eachgk is non-negative, and this sequence of functions is monotonically increasing, but its limit ask → ∞ is1Q, which is not Riemann integrable.

Unsuitability for most domains and functions

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The Riemann integral in its original form is defined over a closed bounded interval of the real line for real functions that are defined on the interval's entirety and continuousalmost everywhere within it. It can, however, be extended to the complex plane, Euclidean space or unions of integrable regions with similar limitations. Animproper Riemann integral, meanwhile, can integrate a function by taking limits on unbounded intervals or at points at which the function is not defined so long as the proper integral approaches a limit as the region of proper integration approaches the desired region of improper integration. While improper integration is an advantage of the Riemann integral, many Lebesgue-integrable functions are not well behaved enough or are defined on domains that are too irregular to be suitable for proper Riemann integration.

Integrating on structures other than Euclidean space

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The Riemann integral is inextricably linked to the order structure of the real line.

Basic theorems of the Lebesgue integral

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Two functions are said to be equalalmost everywhere (f =a.e. g{\displaystyle f\ {\stackrel {\text{a.e.}}{=}}\ g} for short) if{x∣f(x)≠g(x)}{\displaystyle \{x\mid f(x)\neq g(x)\}} is a subset of anull set. Measurability of the set{x∣f(x)≠g(x)}{\displaystyle \{x\mid f(x)\neq g(x)\}} isnot required.

The following theorems are proved in most textbooks on measure theory and Lebesgue integration.[9]

Necessary and sufficient conditions for the interchange of limits and integrals were proved by Cafiero,[10][11][12][13] generalizing earlier work of Renato Caccioppoli, Vladimir Dubrovskii, and Gaetano Fichera.[14]

Alternative formulations

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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 theDaniell integral.

There is also an alternative approach to developing the theory of integration via methods offunctional analysis. The Riemann integral exists for any continuous functionf ofcompactsupport defined onRn (or a fixed open subset). Integrals of more general functions can be built starting from these integrals.

LetCc be the space of all real-valued compactly supported continuous functions ofR. Define a norm onCc by‖f‖=∫|f(x)|dx.{\displaystyle \left\|f\right\|=\int |f(x)|\,dx.}

ThenCc is a normed vector space (and in particular, it is a metric space.) All metric spaces haveHausdorff completions, so letL1 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 onCc, which is dense inL1. Hence∫ has a unique extension to all ofL1. This integral is precisely the Lebesgue integral.

More generally, when the measure space on which the functions are defined is also alocally compacttopological space (as is the case with the real numbersR), measures compatible with the topology in a suitable sense (Radon measures, of which the Lebesgue measure is an example) an integral with respect to them can be defined in the same manner, starting from the integrals ofcontinuous functions withcompact support. More precisely, the compactly supported functions form avector space that carries a naturaltopology, and a (Radon) measure is defined as a continuouslinear functional on this space. The value of a measure at a compactly supported function is then also by definition the integral of the function. One then proceeds to expand the measure (the integral) to more general functions by continuity, and defines the measure of a set as the integral of its indicator function. This is the approach taken by Nicolas Bourbaki[15] and a certain number of other authors. For details seeRadon measures.

Limitations of Lebesgue integral

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The main purpose of the Lebesgue integral is to provide an integral notion where limits of integrals hold under mild assumptions. There is no guarantee that every function is Lebesgue integrable. But it may happen thatimproper integrals exist for functions that are not Lebesgue integrable. One example would be thesinc function:sinc⁡(x)=sin⁡(x)x{\displaystyle \operatorname {sinc} (x)={\frac {\sin(x)}{x}}}over the entire real line. This function is not Lebesgue integrable, as∫−∞∞|sin⁡(x)x|dx=∞.{\displaystyle \int _{-\infty }^{\infty }\left|{\frac {\sin(x)}{x}}\right|dx=\infty .}On the other hand,∫−∞∞sin⁡(x)xdx{\textstyle \int _{-\infty }^{\infty }{\frac {\sin(x)}{x}}\,dx} exists as an improper integral and can be computed to be finite; it is twice theDirichlet integral and equal toπ.

See also

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Notes

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  1. ^This approach can be found in most treatments of measure and integration, such asRoyden (1988).
  2. ^Lemma 1 of page 76 of the second edition of Royden, Real Analysis.
  3. ^ However,L1 is not "the space of Lebesgue integrable functions" but rather the space of equivalence classes of functions.
  4. ^Lieb & Loss 2001
  5. ^Iff ∗ is infinite at an interior point of the domain, then the integral must be taken to be infinity. Otherwisef ∗ is finite everywhere on(0, +∞), and hence bounded on every finite interval[a,b], wherea > 0. Therefore the improper Riemann integral (whether finite or infinite) is well defined.
  6. ^Equivalently, one could have definedf∗(t) = μ({x∈E∣f(x)≥t}),{\displaystyle f^{*}(t)\ =\ \mu \left(\{x\in E\mid f(x)\geq t\}\right),} sinceμ({x∈E∣f(x)≥t})=μ({x∈E∣f(x)>t}){\displaystyle \mu \left(\{x\in E\mid f(x)\geq t\}\right)=\mu \left(\{x\in E\mid f(x)>t\}\right)} for almost all⁠t{\displaystyle t}⁠.
  7. ^Rudin 1966
  8. ^Whitney 1957
  9. ^Folland 1999
  10. ^Cafiero, F. (1953), "Sul passaggio al limite sotto il segno d'integrale per successioni d'integrali di Stieltjes-Lebesgue negli spazi astratti, con masse variabili con gli integrandi [On the passage to the limit under the integral symbol for sequences of Stieltjes–Lebesgue integrals in abstract spaces, with masses varying jointly with integrands]" (Italian), Rendiconti del Seminario Matematico della Università di Padova, 22: 223–245, MR0057951, Zbl 0052.05003.
  11. ^Cafiero, F. (1959), Misura e integrazione [Measure and integration] (Italian), Monografie matematiche del Consiglio Nazionale delle Ricerche 5, Roma: Edizioni Cremonese, pp. VII+451, MR0215954, Zbl 0171.01503.
  12. ^Letta, G. (2013), Argomenti scelti di Teoria della Misura [Selected topics in Measure Theory], (in Italian) Quaderni dell'Unione Matematica Italiana 54, Bologna: Unione Matematica Italiana, pp. XI+183, ISBN 88-371-1880-5, Zbl 1326.28001. Ch. VIII, pp. 110–128
  13. ^Daniele Tampieri (https://mathoverflow.net/users/113756/daniele-tampieri), Do you know important theorems that remain unknown?, URL (version: 2021-12-31):https://mathoverflow.net/q/296839
  14. ^Fichera, G. (1943), "Intorno al passaggio al limite sotto il segno d'integrale" [On the passage to the limit under the integral symbol] (Italian), Portugaliae Mathematica, 4 (1): 1–20, MR0009192, Zbl 0063.01364.
  15. ^Bourbaki 2004.

References

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  • Bartle, Robert G. (1995).The elements of integration and Lebesgue measure. Wiley Classics Library. New York: John Wiley & Sons Inc. xii+179.ISBN 0-471-04222-6.MR 1312157.
  • Bauer, Heinz (2001).Measure and Integration Theory. De Gruyter Studies in Mathematics 26. Berlin: De Gruyter. 236.ISBN 978-3-11-016719-1.
  • Bourbaki, Nicolas (2004).Integration. I. Chapters 1–6. Translated from the 1959, 1965 and 1967 French originals by Sterling K. Berberian. Elements of Mathematics (Berlin). Berlin: Springer-Verlag. xvi+472.ISBN 3-540-41129-1.MR 2018901.
  • Dudley, Richard M. (1989).Real analysis and probability. The Wadsworth & Brooks/Cole Mathematics Series. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software. xii+436.ISBN 0-534-10050-3.MR 0982264. Very thorough treatment, particularly for probabilists with good notes and historical references.
  • Folland, Gerald B. (1999).Real analysis: Modern techniques and their applications. Pure and Applied Mathematics (New York) (Second ed.). New York: John Wiley & Sons Inc. xvi+386.ISBN 0-471-31716-0.MR 1681462.
  • Halmos, Paul R. (1950).Measure Theory. New York, N. Y.: D. Van Nostrand Company, Inc. pp. xi+304.MR 0033869. A classic, though somewhat dated presentation.
  • "Lebesgue integral",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
  • Lebesgue, Henri (1904),Leçons sur l'intégration et la recherche des fonctions primitives, Paris: Gauthier-Villars
  • Lebesgue, Henri (1972).Oeuvres scientifiques (en cinq volumes) (in French). Geneva: Institut de Mathématiques de l'Université de Genève. p. 405.MR 0389523.
  • Lieb, Elliott;Loss, Michael (2001).Analysis.Graduate Studies in Mathematics. Vol. 14 (2nd ed.).American Mathematical Society.ISBN 978-0821827833.
  • Loomis, Lynn H. (1953).An introduction to abstract harmonic analysis. Toronto-New York-London: D. Van Nostrand Company, Inc. pp. x+190.MR 0054173. Includes a presentation of the Daniell integral.
  • Marsden (1974),Elementary classical analysis, W. H. Freeman
  • Munroe, M. E. (1953).Introduction to measure and integration. Cambridge, Mass.: Addison-Wesley Publishing Company Inc. pp. x+310.MR 0053186. Good treatment of the theory of outer measures.
  • Royden, H. L. (1988).Real analysis (Third ed.). New York: Macmillan Publishing Company. pp. xx+444.ISBN 0-02-404151-3.MR 1013117.
  • Rudin, Walter (1976).Principles of mathematical analysis. International Series in Pure and Applied Mathematics (Third ed.). New York: McGraw-Hill Book Co. pp. x+342.MR 0385023. Known asLittle Rudin, contains the basics of the Lebesgue theory, but does not treat material such asFubini's theorem.
  • Rudin, Walter (1966).Real and complex analysis. New York: McGraw-Hill Book Co. pp. xi+412.MR 0210528. Known asBig 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.
  • Saks, Stanisław (1937).Theory of the Integral. Monografie Matematyczne. Vol. 7 (2nd ed.).Warsaw-Lwów: G.E. Stechert & Co.JFM 63.0183.05.Zbl 0017.30004.. English translation byLaurence Chisholm Young, with two additional notes byStefan Banach.
  • Shilov, G. E.; Gurevich, B. L. (1977).Integral, measure and derivative: a unified approach. Translated from the Russian and edited by Richard A. Silverman. Dover Books on Advanced Mathematics. New York: Dover Publications Inc. xiv+233.ISBN 0-486-63519-8.MR 0466463. Emphasizes theDaniell integral.
  • Siegmund-Schultze, Reinhard (2008), "Henri Lebesgue", in Timothy Gowers; June Barrow-Green; Imre Leader (eds.),Princeton Companion to Mathematics, Princeton University Press
  • Teschl, Gerald.Topics in Real and Functional Analysis. (lecture notes).
  • Whitney, H. (1957),Geometric Integration Theory, Princeton Mathematical Series, vol. 21, Princeton, NJ and London:Princeton University Press andOxford University Press, pp. XV+387,MR 0087148,Zbl 0083.28204
  • Yeh, James (2006).Real Analysis: Theory of Measure and Integral 2nd. Edition Paperback. Singapore: World Scientific Publishing Company Pte. Ltd. p. 760.ISBN 978-981-256-6.
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