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| In [[mathematics]], the '''Fatou–Lebesgue theorem''' establishes a chain of [[inequality (mathematics)|inequalities]] relating the [[integral]]s (in the sense of [[Lebesgue integration|Lebesgue]]) of the [[limit superior and limit inferior|limit inferior]] and the [[limit superior and limit inferior|limit superior]] of a [[sequence]] of [[function (mathematics)|function]]s to the limit inferior and the limit superior of integrals of these functions. The theorem is named after [[Pierre Fatou]] and [[Henri Léon Lebesgue]].
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| If the sequence of functions converges [[pointwise convergence|pointwise]], the inequalities turn into [[equality (mathematics)|equalities]] and the theorem reduces to Lebesgue's [[dominated convergence theorem]].
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| ==Statement of the theorem==
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| Let ''f''<sub>1</sub>, ''f''<sub>2</sub>, ... denote a sequence of [[real number|real]]-valued [[measurable function|measurable]] functions defined on a [[measure space]] (''S'',''Σ'',''μ''). If there exists a Lebesgue-integrable function ''g'' on ''S'' which dominates the sequence in absolute value, meaning that |''f''<sub>''n''</sub>| ≤ ''g'' for all [[natural number]]s ''n'', then all ''f''<sub>''n''</sub> as well as the limit inferior and the limit superior of the ''f''<sub>''n''</sub> are integrable and
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| :<math>
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| \int_S \liminf_{n\to\infty} f_n\,d\mu
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| \le \liminf_{n\to\infty} \int_S f_n\,d\mu
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| \le \limsup_{n\to\infty} \int_S f_n\,d\mu
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| \le \int_S \limsup_{n\to\infty} f_n\,d\mu\,.
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| </math>
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| Here the limit inferior and the limit superior of the ''f''<sub>''n''</sub> are taken pointwise. The integral of the absolute value of these limiting functions is bounded above by the integral of ''g''.
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| Since the middle inequality (for sequences of real numbers) is always true, the directions of the other inequalities are easy to remember.
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| ==Proof==
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| All ''f''<sub>''n''</sub> as well as the limit inferior and the limit superior of the ''f''<sub>''n''</sub> are measurable and dominated in absolute value by ''g'', hence integrable.
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| The first inequality follows by applying [[Fatou's lemma]] to the non-negative functions ''f''<sub>''n''</sub> + ''g'' and using the [[Lebesgue_integral#Basic_theorems_of_the_Lebesgue_integral|linearity of the Lebesgue integral]]. The last inequality is the [[Fatou's_lemma#Reverse_Fatou_lemma|reverse Fatou lemma]].
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| Since ''g'' also dominates the limit superior of the |''f''<sub>''n''</sub>|,
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| :<math>0\le\biggl|\int_S \liminf_{n\to\infty} f_n\,d\mu\biggr|
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| \le\int_S \Bigl|\liminf_{n\to\infty} f_n\Bigr|\,d\mu
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| \le\int_S \limsup_{n\to\infty} |f_n|\,d\mu
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| \le\int_S g\,d\mu</math>
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| by the [[Lebesgue_integral#Basic_theorems_of_the_Lebesgue_integral|monotonicity of the Lebesgue integral]]. The same estimates hold for the limit superior of the ''f''<sub>''n''</sub>.
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| == References ==
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| *[http://www.mat.univie.ac.at/~gerald/ftp/book-fa/index.html Topics in Real and Functional Analysis] by [[Gerald Teschl]], University of Vienna.
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| ==External links==
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| *{{planetmath reference|id=3679|title=Fatou-Lebesgue theorem}}
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| {{DEFAULTSORT:Fatou-Lebesgue theorem}}
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| [[Category:Theorems in real analysis]]
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| [[Category:Theorems in measure theory]]
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| [[Category:Articles containing proofs]]
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Nice to satisfy you, my title is Refugia. California is exactly where I've always been living and I adore every day residing here. Bookkeeping is what I do. To gather coins is a factor that I'm totally addicted to.
Feel free to visit my blog post; at home std test (mouse click the up coming post)