Henri Lebesgue (6356 bytes)
1: '''Henri Léon Lebesgue''' ([[June 28]], [[1875]] - [[July 26]], [[1941]]...
2: ...ished in his dissertation, ''[[A summary of Henri Lebesgue's dissertation|Intégral, longueur, aire]]'' ("Int...
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: ...region bounded by the graph of that function. '''Lebesgue integration''' is a mathematical [[theory]] that ...
3: The Lebesgue integral plays an important role in the branch of...
5: The Lebesgue integral is named for [[Henri Lebesgue]] ([[1875]]-[[1941]]). The pronunciation of his n...
16: ... Riemann definition, which is the main reason the Lebesgue integral is better behaved.
17: The Lebesgue definition also makes it possible to calculate in...

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 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: ...''¹ integrable, that is to say if the Lebesgue integral of |''f''| is finite, then
27: {{planetmath|id=3571|title=Riemann-Lebesgue lemma}}

Lebesgue-Stieltjes integration (6144 bytes)
1: ...earn more about [[integral]]s, particularly the [[Lebesgue integral]], and [[measure theory]].''
4: ...mann-Stieltjes integral|Riemann-Stieltjes]] and [[Lebesgue]] [[integral|integration]], preserving the many a...
6: ...d [[Thomas Joannes Stieltjes]], are also known as Lebesgue-Radon integrals or just Radon integrals, after [[...
10: In order to define the Lebesgue-Stieltjes integral, we will begin by associating ...
14: ... construction|construction]] of the corresponding Lebesgue integral. If (&Omega;, '''''F''''', &mu;<sub>''w...

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

Almost everywhere (1778 bytes)
1: ...sed for properties of the [[real number]]s, the [[Lebesgue measure]] is assumed unless otherwise stated.
6: ...R''' <tt>-></tt> '''R''' is a [[Lebesgue integral|Lebesgue integrable]] function and ''f''(''x'') &ge; 0 alm...
9: * If ''f'' : '''R''' &rarr; '''R''' is Lebesgue measurable and
11: ...n ''f'') such that, if ''x'' is not in ''E'', the Lebesgue mean
13: ...'' almost everywhere. The set ''E'' is called the Lebesgue set of ''f''.

Integrable function (1235 bytes)
1: ...," etc. Below we will only examine the concept of Lebesgue integrability.
3: ...>-</sup>'' are measurable functions with finite [[Lebesgue integral]]. Let

Stieltjes integral (1003 bytes)
9: ...ion, [[Darboux integral|Darboux integration]], or Lebesgue integration.
13: ...ntegration]], or [[Lebesgue-Stieltjes integration|Lebesgue-Stieltjes integration]]. Function ''f'' is calle...

Null set (3631 bytes)
15: ...ually understood that the measure being used is [[Lebesgue measure]].
26: === In Lebesgue measure ===
28: For Lebesgue measure on <b>R</b>^<i>n</i>, all [[sin...
35: ...any [[topological manifold]], even if there is no Lebesgue measure there.
39: ...e in the definition of the [[Lebesgue integration|Lebesgue integral]]: if functions <i>f</i> and <i>g</i> ar...

Dominated convergence theorem (1458 bytes)
1: In [[mathematics]], [[Henri Lebesgue]]'s '''dominated convergence theorem''' states th...
13: ...'' may be dispensed with. In [[Lebesgue integral|Lebesgue's theory of integration]], one calls a function "...

Probability density function (3277 bytes)
3: ...'(''x'') is a non-negative [[Lebesgue integration|Lebesgue-integrable]] function '''R''' &rarr; '''R''' such...
7: ... of ''f'' must be 1. Conversely, any non-negative Lebesgue-integrable function with total integral 1 is the ...
27: ...they differ only on a set of [[Lebesgue measure | Lebesgue]] [[measure zero]].

Borel measure (659 bytes)
4: ... is preferred: every Borel measurable set is also Lebesgue measurable, and the measures of the set agree.

Riemann-Stieltjes integral (3468 bytes)
11: ...ion that is more general, see [[Lebesgue integral|Lebesgue integration]].
31: ...[probability density function]] with respect to [[Lebesgue measure]], and ''f'' is any function for which th...
35: ...ve a probability density function with respect to Lebesgue measure. In particular, it does not work if the ...
44: * [[Lebesgue integral]]

Singular (1230 bytes)
5: ...sure &mu;", and that means that &mu; replaces the Lebesgue measure above.

Fourier inversion theorem (3088 bytes)
13: ...theorem assumes that ''f'' is [[Lebesgue integral|Lebesgue-integrable]], i.e., the integral of its absolute ...
17: ...at case, the Fourier transform is not necessarily Lebesgue-integrable; it may be only "conditionally integra...
25: rather than a Lebesgue integral.

Monotone convergence theorem (1228 bytes)
8: 3) If f_k are non-negative [[Lebesgue integral|measurable]] real-valued functions with ...
10: ...es the previous one. It is sometimes called the ''Lebesgue monotone convergence theorem''; and is probably t...

Henstock-Kurzweil integral (4166 bytes)
1: ...ral]] which proves to be more powerful than the [[Lebesgue integral]].
3: ...egral]] (the oldest and simplest definition) or [[Lebesgue integral]] (the most common).''
9: ...lity]]). In effect, the definitions of Denjoy and Lebesgue agree completely on positive functions.
13: ...particular a non-trivial corrolary applies to the Lebesgue integral: if a function ''f'' is differentiable e...
55: * Russell A. Gordon, ''The integrals of Lebesgue, Denjoy, Perron, and Henstock'', Graduate Studies...

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