The integral of a positive function can be interpreted as the area under a curve.

In mathematics, the integral of a nonnegative function can be regarded in the simplest case as the area between the graph of that function and the x-axis. Lebesgue integration is a mathematical construction that extends the integral 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 nonnegative 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 by polygons. However, as the need to consider more irregular functions arose (for example, as a result of the limiting processes of mathematical analysis and the mathematical theory of probability) it became clear that more careful approximation techniques would be needed in order to define a suitable integral. Image File history File links Integral-area-under-curve. ... Image File history File links Integral-area-under-curve. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... This article is about the concept of integrals in calculus. ... Analysis has its beginnings in the rigorous formulation of calculus. ... Probability theory is the mathematical study of probability. ...

The Lebesgue integral plays an important role in the branch of mathematics called real analysis and in many other fields in the mathematical sciences. Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ...

The Lebesgue integral is named for Henri Lebesgue (1875-1941). His last name is pronounced as [ləˈbɛg], which may be approximated in English as luh beg. Henri Lebesgue Henri Léon Lebesgue (June 28, 1875, Beauvais – July 26, 1941, Paris) was a French mathematician, most famous for his theory of integration. ... 1875 (MDCCCLXXV) was a common year starting on Friday (see link for calendar). ... For other uses, see 1941 (disambiguation). ...

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 to Lebesgue measure. In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ... In mathematics, the real line is simply the set of real numbers. ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ...

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. In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

As part of a general movement toward rigour in mathematics in the nineteenth century, attempts were made to put the integral calculus on a firm foundation. The Riemann integral, proposed by Bernhard Riemann (1826-1866), is a broadly successful attempt to provide such a foundation for the integral. Riemann's definition starts with the construction of a sequence of easily-calculated integrals which converge to the integral of a given function. This definition is successful in the sense that it gives the expected answer for many already-solved problems, and gives useful results for many other problems. Look up Rigour in Wiktionary, the free dictionary. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ... Bernhard Riemann. ... The oldest surviving photograph, Nicéphore Niépce, circa 1826 1826 (MDCCCXXVI) was a common year starting on Sunday (see link for calendar) of the Gregorian calendar (or a common year starting on Tuesday of the 12-day-slower Julian calendar). ... 1866 (MDCCCLXVI) is a common year starting on Monday of the Gregorian calendar or a common year starting on Wednesday of the 12-day-slower Julian calendar. ...

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 transforms and other topics. The Lebesgue integral is better able to describe how and when it is possible to take limits under the integral sign. The Lebesgue definition considers a different class of easily-calculated integrals than the Riemann definition, which is the main reason the Lebesgue integral is better behaved. The Lebesgue definition also makes it possible to calculate integrals for a broader class of functions. For example, the Dirichlet function, which is 0 where its argument is irrational and 1 otherwise, has a Lebesgue integral, but it does not have a Riemann integral. The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. ... In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers. ...

In the next section, we discuss the technical definition of the Lebesgue 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:

1. A theory of measurable sets and measures on these sets.
2. 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 in set theory (see non-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. In mathematics, a measure is a function that assigns a number, e. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, a non-measurable set is a set whose structure is so complicated it sheds light on the very notion of length, area or volume. ...

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. In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ...

Integration

We will work in the following abstract setup: μ is a (non-negative) measure on a σ-algebra X of subsets of E. For example, E can be Euclidean n-space Rⁿ or some Lebesgue measurable subset of it, X will be the σ-algebra of all Lebesgue measurable subsets of E, and μ will be the Lebesgue measure. In the mathematical theory of probability μ will be a probability measure on a probability space E. In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ... Probability is the likelihood that something is the case or will happen. ...

In Lebesgue's theory, integrals are limited to a class of functions called measurable functions. A function f is measurable if the pre-image of every closed interval is in X: In mathematics, measurable functions are well-behaved functions between measurable spaces. ...

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: In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space X is a σ-algebra of subsets of X associated to the topology of X. In the mathematics literature, there are at least two inequivalent definitions of this σ-algebra: The minimal σ-algebra containing the open sets. ...

are measurable if the original sequence {f_k}, where k ∈ N, consists of measurable functions.

We build up an integral

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

Indicator functions: To assign a value to the integral of the indicator function of a measurable set S consistent with the given measure μ, the only reasonable choice is to set: In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ...

Simple functions: We extend by linearity to the linear span of indicator functions: In the mathematical subfield of linear algebra, the linear span, also called the linear hull, of a set of vectors in a vector space is the intersection of all subspaces containing that set. ...

where the sum is finite and the coefficients a_k are real numbers. Such a finite linear 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. In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In mathematics, especially in mathematical analysis, a simple function is a measurable function whose range is finite. ...

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 the extended real number line. We define The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ...

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 not hard 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

where

Note that both f⁺ and f⁻ are non-negative functions. Also note that

If

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

and it makes sense to define

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. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

Intuitive interpretation

Illustration of a Riemann integral (blue) and a Lebesgue integral (red)

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). Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

The Riemann-Darboux approach: Divide the base of the mountain into a grid of 1 meter squares (a cadaster, 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. Cadastral map of village Pielnia, 1852, Galicia, Austrian Empire. ...

The Lebesgue approach: Draw a contour 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. Example of a topographic map with contour lines Topographic maps, also called contour maps, topo maps or topo quads (for quadrangles), are maps that show topography, or land contours, by means of contour lines. ...

Folland ^[1] 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".

Example

Consider the indicator function of the rational numbers, 1_Q. This function is nowhere continuous. In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ... In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. ...

• 1_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 upper Darboux sums will all be one, and the lower Darboux sums will all be zero.

• 1_Q is Lebesgue-integrable on [0,1]: Indeed it is the indicator function of the rationals so by definition

since Q is countable.

In real analysis, a branch of mathematics, the Darboux integral is one possible definition of the integral of a function. ...

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. If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ...

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 The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...

and

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, the Monotone convergence theorem fails. To see why, let {a_k} be an enumeration of all the rational numbers in [0,1] (they are countable so this can be done.) Then let Monotone convergence theorem, in mathematics, may refer to several theorems, all of which are concerned with a monotonic function in one way or another: Monotonic function refers to the convergence of an infinite series that is monotonic Dominated convergence theorem refers to Lebesgues monotone convergence theorem Categories: | | ... In mathematics the term countable set is used to describe the size of a set, e. ...

Then let

The function f_k is zero everywhere except on a finite set of points, hence its Riemann integral is zero. The sequence f_k is also clearly non-negative and 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. The simplest extension is to define

whenever the limit exists. However, this breaks the desirable property of translation invariance: if f and g are zero outside some interval [a, b] and are Riemann integrable, and if f(x) = g(x + y) for some y, then ∫ f = ∫ g. With this definition of the improper integral (this definition is sometimes called the improper Cauchy principal value about zero), the functions f(x) = (1 if x > 0, −1 otherwise) and g(x) = (1 if x > 1, −1 otherwise) are translations of one another, but their improper integrals are different. It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ... In mathematics, the Cauchy principal value of certain improper integrals is defined as either the finite number where b is a point at which the behavior of the function f is such that for any a < b and for any c > b (one sign is + and the other is −). or...

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 In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ... ↔ ⇔ ≡ logical symbols representing iff. ...

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

• 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 In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

Monotonicity: If f ≤ g, then In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ...

Monotone convergence theorem: Suppose {f_k}_{k ∈ N} is a sequence of non-negative measurable functions such that In mathematics, Lebesgues dominated convergence theorem states that if a sequence { fn : n = 1, 2, 3, ... } of real-valued measurable functions on a measure space S converges almost everywhere, and is dominated (explained below) by some nonnegative function g in , then It is proven using Fatous lemma. ...

Then

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

Fatou's lemma: If {f_k}_{k ∈ N} is a sequence of non-negative measurable functions, then Fatous lemma establishes an inequality relating the integral (in the sense of Lebesgue) of the limit inferior of a sequence of functions to the limit inferior of the sequence of integrals of the functions. ...

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

Dominated convergence theorem: If {f_k}_{k ∈ N} is a sequence of measurable functions with pointwise limit f, and if there is a Lebesgue integrable function g such that |f_k| ≤ g for all k, then f is Lebesgue integrable and In mathematics, Lebesgues dominated convergence theorem states that if a sequence { fn : n = 1, 2, 3, ... } of real-valued measurable functions on a measure space S converges almost everywhere, and is dominated (explained below) by some nonnegative function g in , then It is proven using Fatous lemma. ...

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 ∈ N} be a non-decreasing sequence of non-negative measurable functions and put

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

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

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

Therefore, it suffices to prove that for each k ∈ N,

We will show that if g is a simple function and

almost everywhere, then

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 ∈ N} is a nondecreasing sequence of measurable functions on E such that

for almost all x ∈ A. Then

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

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

By assumption,

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

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. The construction of the Lebesgue Integral is built on top of measure theory. ...

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 Rⁿ (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 Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...

Then C_c is a normed vector space (and in particular, it is a metric space.) All metric spaces have Hausdorff completions, so let L¹ 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 uniformly continuous functional with respect to the norm on C_c, which is dense in L¹. Hence ∫ has a unique extension to all of L¹. This integral is precisely the Lebesgue integral. In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x effect small changes in the output f(x) (continuity), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but...

This approach can be generalised to build the theory of integration with respect to Radon measures on locally compact spaces. It is the approach adopted by Bourbaki (2004); for more details see Radon measures on locally compact spaces. In mathematics, a Radon measure on a Hausdorff topological space X is a measure on the σ-algebra of Borel sets of X that is locally finite and inner regular. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ... In mathematics, a Radon measure on a Hausdorff topological space X is a measure on the σ-algebra of Borel sets of X that is locally finite and inner regular. ...

See also

• null set
• integration
• measure
• sigma-algebra
• Lebesgue measure
• Lebesgue space
• Lebesgue-Stieltjes integration
• Henstock-Kurzweil integral

In measure theory, a null set is a set that is negligible for the purposes of the measure in question. ... This article is about the concept of integrals in calculus. ... In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ... In mathematics, a &#963;-algebra (or &#963;-field) X over a set S is a family of subsets of S which is closed under countable set operations; &#963;-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ... The title given to this article is incorrect due to technical limitations. ... In measure-theoretic analysis and related branches of mathematics, Lebesgue-Stieltjes integration generalizes Riemann-Stieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure-theoretic framework. ... In mathematics, the Henstock-Kurzweil integral, also known as the Denjoy integral (pronounce Denjua) and the Perron integral, is a possible definition of the integral of a function. ...

Notes

1. ^ Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 1984, p. 56.

References

• Bartle, Robert G. (1995). The elements of integration and Lebesgue measure, Wiley Classics Library. New York: John Wiley & Sons Inc., pp. xii+179. ISBN 0-471-04222-6.  MR1312157

• 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, pp. xvi+472. ISBN 3-540-41129-1.  MR2018901

• Dudley, Richard M. (1989). Real analysis and probability, The Wadsworth &ammp; Brooks/Cole Mathematics Series. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, pp. xii+436. ISBN 0-534-10050-3.  MR982264 Very thorough treatment, particularly for probabilists with good notes and historical references.

• Folland, Gerald B. (1999). Real analysis: Modern techniques and their applications, Second edition, Pure and Applied Mathematics (New York), New York: John Wiley & Sons Inc., pp. xvi+386. ISBN 0-471-31716-0.  MR1681462

• Halmos, Paul R. (1950). Measure Theory. New York, N. Y.: D. Van Nostrand Company, Inc., pp. xi+304.  MR0033869 A classic, though somewhat dated presentation.

• Lebesgue, Henri (1972). Oeuvres scientifiques (en cinq volumes) (in French). Geneva: Institut de Mathématiques de l'Université de Genève, pp. 405.  MR0389523

• Loomis, Lynn H. (1953). An introduction to abstract harmonic analysis. Toronto-New York-London: D. Van Nostrand Company, Inc., pp. x+190.  MR0054173 Includes a presentation of the Daniell integral.

• Munroe, M. E. (1953). Introduction to measure and integration. Cambridge, Mass.: Addison-Wesley Publishing Company Inc., pp. x+310.  MR0053186 Good treatment of the theory of outer measures.

• Royden, H. L. (1988). Real analysis, Third edition, New York: Macmillan Publishing Company, pp. xx+444. ISBN 0-02-404151-3.  MR1013117

• Rudin, Walter (1976). Principles of mathematical analysis, Third edition, International Series in Pure and Applied Mathematics, New York: McGraw-Hill Book Co., pp. x+342.  MR0385023 Known as Little Rudin, contains the basics of the Lebesgue theory, but does not treat material such as Fubini's theorem.

• Rudin, Walter (1966). Real and complex analysis. New York: McGraw-Hill Book Co., pp. xi+412.  MR0210528 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.

• 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 (in English). Dover Publications Inc., pp. xiv+233. ISBN 0-486-63519-8.  MR0466463 Emphasizes the Daniell integral.

Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... Nicolas Bourbaki is the collective allonym under which a group of (mainly French) 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... Paul Halmos Paul Richard Halmos (March 3, 1916 — October 2, 2006) was a Hungarian-born American mathematician who wrote on probability theory, statistics, operator theory, ergodic theory, functional analysis (in particular, Hilbert spaces), and mathematical logic. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... Henri Lebesgue Henri Léon Lebesgue (June 28, 1875, Beauvais – July 26, 1941, Paris) was a French mathematician, most famous for his theory of integration. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... Walter Rudin Walter Rudin is an American mathematician, formerly a professor of mathematics at the University of Wisconsin, Madison. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... It has been suggested that A counterexample related to Fubinis theorem be merged into this article or section. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... The construction of the Lebesgue Integral is built on top of measure theory. ... This article is about the concept of integrals in calculus. ... If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ... In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ... The construction of the Lebesgue Integral is built on top of measure theory. ... If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ... In mathematics, the Henstock-Kurzweil integral, also known as the Denjoy integral (pronounce Denjua) and the Perron integral, is a possible definition of the integral of a function. ... In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ... If you are having difficulty understanding this article, you might want to first learn more about integrals, particularly the Lebesgue integral, and measure theory. ... In mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. ...