Integrasyong Lebesgue: Pagkakaiba sa mga binago
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Linya 20:
\end{cases}
\end{align}</math>
==Konstruksiyon==
Ito ay nagsisimula sa isang [[espasyong sukat]] (''E'', ''X'', μ) kung saan ''E'' ay isang hanay, ang ''X'' ay isang [[sigma-algebra|σ-alhebra]] ng mga subhanay ng ''E'' at ang ''μ'' ay isang hindi negatibong sukat sa E na inilalarawan sa mga hanay ng X. Halimbawa, ang E ay maaaring [[espasyong Euclidean|epasyong-''n'' na Euclidean]] na '''R'''<sup>''n''</sup> o isang [[sukat Lebesgue|masusukat na Lebesgue]] na subhanay nito, ang ''X'' ang [[σ-alhebra]] ng ''E'', at ang ''μ'' ang [[sukat Lebesgue]]. Sa teoriya ng probabilidad, ang pag-aaral ay nililimita sa sukat ng [[probabilidad]] ''μ'' na sumasapat sa<math>\mu(E) = 1</math>. Sa teoriya ng Lebesgue, ang mga integral ay nilalarawan para sa isang klase ng mga punsiyong tinatawag na mga [[masusukat na punsiyon]]. Ang isang ''ƒ'' ay masusukat kung ang [[pre-imahe]] ng bawat interbal ng anyong <math>(t,\infty)</math> ay nasa ''X'':
:<math> \{x\,\mid\,f(x) > t\} \in X\quad \text{for all}\ t\in\mathbb{R}. </math>
Maipapakita na ito ay katumbas ng pag-aatas ng pre-imahe ng anumang subhanay na [[alhebrang Borel|Borel]] ng '''R''' na nasa ''X''. Ang hanay ng mga masusukat na punsiyon ay sarado sa ilalim ng mga operasyong alhebraiko ngunit sa mas mahalaga, ang klase ay sarado sa ilalim ng iba't ibang mga uri ng [[sa puntong mga hangganang sekwensiyal]]:
: <math> \sup_{k \in \mathbb{N}} f_k, \quad \liminf_{k \in \mathbb{N}} f_k, \quad \limsup_{k \in \mathbb{N}} f_k </math>
ay masusukat kung ang orihinal na sekwensiya(''ƒ''<sub>''k''</sub>)<sub>''k''</sub> kung saan ang ''k'' ∈ '''N''', ay binubuo ng mga masusukat na punsiyon.
==Mga sanggunian==
{{reflist}}
== 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 = xii+179
| isbn = 0-471-04222-6
| nopp = true
| mr = 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 = xvi+472
| isbn = 3-540-41129-1
| nopp = true
| mr = 2018901}}
* {{cite book
| last = Dudley
| first = Richard M.
| title = Real analysis and probability
| series = The Wadsworth & Brooks/Cole Mathematics Series
| publisher = Wadsworth & Brooks/Cole Advanced Books & Software
| location = Pacific Grove, CA
| year = 1989
| pages = xii+436
| isbn = 0-534-10050-3
| nopp = true
| mr = 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
| publisher = John Wiley & Sons Inc.
| location = New York
| year = 1999
| pages = xvi+386
| isbn = 0-471-31716-0
| nopp = true
| mr = 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 = xi+304
| mr = 0033869}} A classic, though somewhat dated presentation.
* {{Cite document
| 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
| postscript = <!--None-->}}
* {{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 = 405
| language = French
| mr = 0389523}}
*{{cite book
|last1=Lieb
|first1=Elliott
|authorlink1=Elliott Lieb
|last2=Loos
|first2=Michael
|title=Analysis
|year=2001
|publisher=AMS Chelsea
|series=Graduate Studies in Mathematics
|isbn=978-0821827833}}
* {{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 = x+190
| mr = 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 = x+310
| mr = 0053186}} Good treatment of the theory of outer measures.
* {{cite book
| last = Royden
| first = H. L.
| title = Real analysis
| edition = Third
| publisher = Macmillan Publishing Company
| location = New York
| year = 1988
| pages = xx+444
| isbn = 0-02-404151-3
| mr = 1013117}}
* {{cite book
| last = Rudin
| first = Walter
| authorlink = Walter Rudin
| title = Principles of mathematical analysis
| edition = Third
| series = International Series in Pure and Applied Mathematics
| publisher = McGraw-Hill Book Co.
| location = New York
| year = 1976
| pages = x+342
| mr = 0385023}} Known as ''Little Rudin'', contains the basics of the Lebesgue theory, but does not treat material such as [[Fubini's theorem]].
* {{cite book
| last = Rudin
| first = Walter
| title = Real and complex analysis
| publisher = McGraw-Hill Book Co.
| location = New York
| year = 1966
| pages = xi+412
| mr = 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 document
| last = Saks
| first = Stanisław
| author-link = Stanislaw Saks
| title = Theory of the Integral
| place = [[Warszawa]]-[[Lwów]]
| publisher = G.E. Stechert & Co.
| year = 1937
| series = [http://matwbn.icm.edu.pl/ksspis.php?wyd=10&jez=pl Monografie Matematyczne]
| volume = 7
| edition = 2nd
| pages = VI+347
| url = http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10&jez=pl
| jfm = 63.0183.05 | zbl = 0017.30004
| postscript = <!--None-->}}. English translation by [[Laurence Chisholm Young]], with two additional notes by [[Stefan Banach]].
* {{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.
| location = New York
| year = 1977
| pages = xiv+233
| isbn = 0-486-63519-8
| nopp = true
| mr = 0466463}} Emphasizes the [[Daniell integral]].
* {{citation|last=Siegmund-Schultze|first=Reinhard|chapter=Henri Lebesgue|title=Princeton Companion to Mathematics|editors=Timothy Gowers, June Barrow-Green, Imre Leader|year=2008|publisher=Princeton University Press}}.
[[Kategorya:Kalkulo]]
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