- Fatou–Lebesgue theorem
In
mathematics , the Fatou–Lebesgue theorem establishes a chain of inequalities relating theintegral s (in the sense of Lebesgue) of the limit inferior and the limit superior of asequence of functions to the limit inferior and the limit superior of integrals of these functions. Thetheorem is named after the Frenchmathematician sPierre Fatou (1878 – 1929) andHenri Léon Lebesgue (1875 – 1941).If the sequence of functions converges pointwise, the inequalities turn into equalities and the theorem reduces to the Lebesgue's
dominated convergence theorem .tatement of the theorem
Let "f"1, "f"2, ... denote a sequence of real-valued 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""n"| ≤ "g" for allnatural number s "n", then all "f""n" as well as the limit inferior and the limit superior of the "f""n" are integrable and:int_S liminf_{n oinfty} f_n,dmule liminf_{n oinfty} int_S f_n,dmule limsup_{n oinfty} int_S f_n,dmule int_S limsup_{n oinfty} f_n,dmu,.Here the limit inferior and the limit superior of the "f""n" are taken pointwise. The integral of the absolute value of these limiting functions is bounded above by the integral of "g".Since the middle inequality (for sequences of real numbers) is always true, the directions of the other inequalities are easy to remember.
Proof
All "f""n" as well as the limit inferior and the limit superior of the "f""n" are measurable and dominated in absolute value by "g", hence integrable.
The first inequality follows by applying
Fatou's lemma to the non-negative functions "f""n" + "g" and using the linearity of the Lebesgue integral. The last inequality is the reverse Fatou lemma.Since "g" also dominates the limit superior of the |"f""n"|,
:0leiggl|int_S liminf_{n oinfty} f_n,dmuiggr
leint_S Bigl|liminf_{n oinfty} f_nBigr|,dmuleint_S limsup_{n oinfty} |f_n|,dmuleint_S g,dmuby the monotonicity of the Lebesgue integral. The same estimates hold for the limit superior of the "f""n".
References
External links
*planetmath reference|id=3679|title=Fatou-Lebesgue theorem
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