Henri Lebesgue (6356 bytes)
1: ...[mathematician]], most famous for his theory of [[integral|integration]].
2: ...gue's dissertation|Intégral, longueur, aire]]'' ("Integral, length, area"), at the University of [[Nancy]] i...
4: Lebesgue's father was a typesetter,
6: Lebesgue himself suffered from poor health throughout his ...
12: Lebesgue married the sister of one of his fellow students,

Lebesgue measure (5519 bytes)
2: ...o, not all subsets of '''R'''^''n'' are Lebesgue measurable. The "strange" behavior of [[non-measu...
6: The Lebesgue measure has the following properties:
8: ...; ... &sdot; ''I''_''n'', then ''A'' is Lebesgue measurable and &lambda;(''A'') = |''I''<sub>1</su...
9: ...nt Lebesgue measurable sets, then ''A'' is itself Lebesgue measurable and &lambda;(''A'') is equal to the su...
10: # If ''A'' is Lebesgue measurable, then so is its complement.

Lebesgue integration (22686 bytes)
1: ...ues would be needed in order to define a suitable integral.
3: The Lebesgue integral plays an important role in the branch of mathemat...
5: The Lebesgue integral is named for [[Henri Lebesgue]] ([[1875]]-[[1941]]). The pronunciation of his n...
9: [[Image:Integral-area-under-curve.png|thumb|The integral is defined as the area under a curve.]]
11: The integral of a function ''f'' between limits ''a'' and ''b'...

Lebesgue-measurable function (252 bytes)
1: In [[mathematics]], a '''Lebesgue-measurable function''' is a real function ''f'' :...
6: is a Lebesgue-measurable set. See [[Lebesgue measure]].

Lebesgue-Stieltjes integration (6144 bytes)
1: ... about [[integral]]s, particularly the [[Lebesgue integral]], and [[measure theory]].''
4: ...s integral|Riemann-Stieltjes]] and [[Lebesgue]] [[integral|integration]], preserving the many advantages of ...
6: ...o known as Lebesgue-Radon integrals or just Radon integrals, after [[Johann Radon]], to whom much of the the...
10: In order to define the Lebesgue-Stieltjes integral, we will begin by associating a measure, &mu;<sub...
14: ...>) is a [[measure space]], then we can define the integral of any [[simple function]] ''s'' = &Sigma;<sub>''...

Lebesgue covering dimension (918 bytes)
1: In [[mathematics]], the '''Lebesgue covering dimension''' of a [[topological space]] ...
5: ...f a finite [[simplicial complex]]; this is the '''Lebesgue covering theorem'''.

Riemann-Lebesgue lemma (1473 bytes)
1: ...t is named after [[Bernhard Riemann]] and [[Henri Lebesgue]]
4: .../sup> integrable, that is to say if the Lebesgue integral of |''f''| is finite, then
27: {{planetmath|id=3571|title=Riemann-Lebesgue lemma}}

Lebesgue's number lemma (382 bytes)
3: In [[topology]], '''Lebesgue's number lemma''' states
8: *[[Henri Lebesgue]]

Integral (11240 bytes)
1: ... integral in [[calculus]]. For other meanings of "integral" see [[integration]].''
5: In [[calculus]], the '''integral''' of a [[function (mathematics)|function]] is a ...
7: Intuitively, the integral of a continuous, positive real-valued function ''...
11: then the integral of ''f'' between ''a'' and ''b'' is the [[measure...
13: ...andard [[long s]] notation for the integral. The integral of the previous paragraph would be written <math>...

Fresnel integral (3791 bytes)
1: ... [[Fresnel diffraction]] phenomena, and are the [[integral]]s defined as follows:
17: ...main|domain]] of [[complex number]]s, the Fresnel integrals can be written using the [[error function]] as f...
22: ...ble (but not [[trivial]]) to evaluate the fresnel integrals in the limits, we have
30: ...e along the real axis will tend to the well known integral
34: ... - along the slope - will evaluate to the Fresnel integrals after some rearangings.

Gaussian integral (3252 bytes)
1: ... Gauss]]) is quickly reducible to the '''Gaussian integral'''
5: ...nverts to polar coordinates. Let the value of the integral be <math>s</math>. Then, by applying [[Fubini's t...
13: (The factor of <math>r</math> that makes the integral straightforward comes from the change of variable...
24: where the integral is understood to be over '''R'''ⁿ.
37: ...t cases), we can ''define'' a Gaussian functional integral in analogy to the finite dimensional case. The is...

Integral type (606 bytes)
1: ...e>, <code>_Bool</code>, and their equivalents are integral types.

Integral geometry (2230 bytes)
2: In [[mathematics]], the term '''''integral geometry''''' in is used in two ways, which, alth...
4: ...what 'random line' means; and expectations become integrals with respect to that measure. (Note for example ...
6: ...space]]s of [[Lie group]]s; and the evaluation of integrals of [[differential form]]s arising.
10: ... of the most interesting theorems in this form of integral geometry is [[Hadwiger's theorem]].
12: ...ase) is seen in a freer light, as the site for an integral transform composed as ''pullback onto the inciden...

Stieltjes integral (1003 bytes)
1: ...vides a direct way of (numerically) defining an [[integral]] of the type
9: ...ion, [[Darboux integral|Darboux integration]], or Lebesgue integration.
13: ...ntegration]], or [[Lebesgue-Stieltjes integration|Lebesgue-Stieltjes integration]]. Function ''f'' is calle...
16: '''See also:''' [[Riemann-Stieltjes integral]].

Improper integral (3908 bytes)
1: ...he reader be familiar with [[antiderivative]]s, [[integral]]s, and [[limit (mathematics)|limit]]s.''
5: ...[[real number]] or &infin; or &minus;&infin;. The integral
13: ...nce it may be interpreted instead as a [[Lebesgue integral]] over the set (0, &infin;).
18: cannot be interpreted as a Lebesgue integral, since
22: This is therefore a "properly" improper integral, whose value is given by

Double integral (2677 bytes)
1: ...etween a '''double integral''' and an '''iterated integral'''. To one who has had an advanced calculus cours...
5: A '''double integral'''
9: ... rather than by integrating twice (see [[Lebesgue integral]]).
19: is an '''iterated integral''', so called because one integrates, and then in...
29: ...rrangement of a conditionally convergent iterated integral]].

Integral equation (2452 bytes)
1: ...etween [[differential equation|differential]] and integral equations, and some problems may be formulated ei...
3: Integral equations are classified in three ways.
4: There are thus eight kinds of integral equations.
6: ...egration is variable, it is called a ''[[Volterra integral equation|Volterra equation]]''.
7: ... and outside the integral sign, it is called an ''integral equation of the second kind''.

Direct integral (16243 bytes)
1: ...direct integrals of [[Hilbert space]]s and direct integrals of [[von Neumann algebra]]s. The concept was in...
3: Results on direct integrals can be viewed as generalizations of results abou...
5: Direct integral theory was also used by [[George Mackey]] in his ...
7: == Direct integrals of Hilbert spaces ==
8: The simplest example of a direct integral are the ''L''² spaces associated to a ...

Integral (philosophy) (2809 bytes)
1: ... [[Don Beck]] and many others now gathered in the Integral Institute.
3: ...unity, holism, dynamic dialecticism, or universal integralism..."
7: ...a coherent view of the topic. In a certain sense, integral approaches are "meta-paradigms," or ways to draw ...
9: * This means that the chief activity of integral cognition is not looking at all of the available ...
13: *[http://www.integralinstitute.com Integral Institute]

Integral politics (1854 bytes)
3: ...reduced to one another. ''All levels'' emphasizes integral politics' developmental perspective, which holds ...
6: * [http://noosphere.cc/integralpolitics.html/The Integral Politics Website]
7: * [http://www.integralinstitute.org/ Integral Institute]
8: * [http://www.integralnaked.org/ Integral Naked]
10: * [http://www.integralworld.net/ Explores theories of everything]

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