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 In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For ...

of a sequence of

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

s to the limit inferior and the limit superior of integrals of these functions. The theorem is named after

Pierre Fatou Pierre Joseph Louis Fatou (28 February 1878 – 9 August 1929) was a French mathematician and astronomer. He is known for major contributions to several branches of analysis. The Fatou lemma and the Fatou set are named after him. Biography P ...

and Henri Léon Lebesgue. If the sequence of functions converges

pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...

, the inequalities turn into equalities and the theorem reduces to Lebesgue's

dominated convergence theorem In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary t ...

Statement of the theorem

Let ''f''₁, ''f''₂, ... denote a sequence of

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

-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_ 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 In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou. Fatou's le ...

to the non-negative functions ''f''_''n'' + ''g'' and using the linearity of the Lebesgue integral. The last inequality is the

reverse Fatou lemma In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou. Fatou's le ...

. 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