mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 Pierre Fatou and

Henri Léon Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his Lebesgue integration, theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an ...

. If the sequence of functions converges pointwise, the inequalities turn into

equalities In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. The equality b ...

and the theorem reduces to Lebesgue's dominated convergence theorem.

Statement of the theorem

Let ''f''₁, ''f''₂, ... denote a sequence of real-valued measurable functions defined on a

measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...

(''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_ f_n\,d\mu
\le \liminf_ \int_S f_n\,d\mu
\le \limsup_ \int_S f_n\,d\mu
\le \int_S \limsup_ f_n\,d\mu\,.

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'', , :

0\le\biggl, \int_S \liminf_ f_n\,d\mu\biggr, 
\le\int_S \Bigl, \liminf_ f_n\Bigr, \,d\mu
\le\int_S \limsup_ , f_n, \,d\mu
\le\int_S g\,d\mu

by the monotonicity of the Lebesgue integral. The same estimates hold for the limit superior of the ''f''_''n''.

References

Topics in Real and Functional Analysis
by Gerald Teschl, University of Vienna.

External links

* {{DEFAULTSORT:Fatou-Lebesgue theorem Theorems in real analysis Theorems in measure theory Articles containing proofs