Fatou–Lebesgue theorem

In mathematics, the Fatou–Lebesgue theorem establishes a chain of inequalities relating the integrals (in the sense of Lebesgue) of the limit inferior and the limit superior of a sequence of functions to the limit inferior and the limit superior of integrals of these functions. The theorem is named after the French mathematicians Pierre Fatou (1878 – 1929) and Henri 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 all natural numbers "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"|,

:
leint_S Bigl|liminf_{n oinfty} f_nBigr|,dmuleint_S limsup_{n oinfty} |f_n|,dmuleint_S g,dmu

by 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