Fatou's lemma

In mathematics, Fatou's lemma establishes an inequality relating the integral (in the sense of Lebesgue) of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after the French mathematician Pierre Fatou (1878 - 1929).

Fatou's lemma can be used to prove the Fatou–Lebesgue theorem and Lebesgue's dominated convergence theorem.

tandard statement of Fatou's lemma

If "f"₁, "f"₂, . . . is a sequence of non-negative measurable functions defined on a measure space ("S","Σ","μ"), then:$int\_S\; liminf\_\{n\; oinfty\}\; f\_n,dmu\; le\; liminf\_\{n\; oinfty\}\; int\_S\; f\_n,dmu,.$

Note: On the left-hand side the limit inferior of the "f"_"n" is taken pointwise. The functions are allowed to attain the value infinity and the integrals may also be infinite.

Proof

Fatou's lemma is here proved using the monotone convergence theorem (it can be proved directly).

Let "f" denote the limit inferior of the "f"_"n". For every natural number "k" define pointwise the function:$g\_k=inf\_\{nge\; k\}f\_n.$Then the sequence "g"₁, "g"₂, . . . is increasing and converges pointwise to "f".For "k" ≤ "n", we have "g"_"k" ≤ "f"_"n", so that:$int\_S\; g\_k,dmuleint\_S\; f\_n,dmu,$hence:$int\_S\; g\_k,dmuleinf\_\{nge\; k\}int\_S\; f\_n,dmu.$Using the monotone convergence theorem, the last inequality, and the definition of the limit inferior, it follows that:$int\_S\; liminf\_\{n\; oinfty\}\; f\_n,dmu=lim\_\{k\; oinfty\}int\_S\; g\_k,dmulelim\_\{k\; oinfty\}\; inf\_\{nge\; k\}int\_S\; f\_n,dmu=liminf\_\{n\; oinfty\}\; int\_S\; f\_n,dmu,.$

Examples for strict inequality

Equip the space $S$ with the Borel σ-algebra and the Lebesgue measure.
* Example for a probability space: Let $S=\; [0,1]$ denote the unit interval. For every natural number $n$ define:
* Example with uniform convergence: Let $S$ denote the set of all real numbers. Define :

These sequences $(f\_n)\_\{ninN\}$ converge on $S$ pointwise (respectively uniform) to the zero function (with zero integral), but every $f\_n$ has integral one.

A counterexample

A suitable assumption concerning the negative parts of the sequence "f"₁, "f"₂, . . . of functions is necessary for Fatou's lemma, as the following example shows. Let "S" denote the half line [0,∞) with the Borel σ-algebra and the Lebesgue measure. For every natural number "n" define:This sequence converges uniformly on "S" to the zero function (with zero integral) and for every "x" ≥ 0 we even have "f_n"("x") = 0 for all "n" > "x" (so for every point "x" the limit 0 is reached in a finite number of steps). However, every function "f_n" has integral −1, hence the inequality in Fatou's lemma fails.

Reverse Fatou lemma

Let "f"₁, "f"₂, . . . be a sequence of extended real-valued measurable functions defined on a measure space ("S","Σ","μ"). If there exists an integrable function "g" on "S" such that "f"_"n" ≤ "g" for all "n", then:$int\_Slimsup\_\{n\; oinfty\}f\_n,dmugelimsup\_\{n\; oinfty\}int\_Sf\_n,dmu.$

Note: Here "g integrable" means that "g" is measurable and that

Proof

Apply Fatou's lemma to the non-negative sequence given by "g" – "f"_"n".

Extensions and variations of Fatou's lemma

Integrable lower bound

Let "f"₁, "f"₂, . . . be a sequence of extended real-valued measurable functions defined on a measure space ("S","Σ","μ"). If there exists a non-negative integrable function "g" on "S" such that "f"_"n" ≥ −"g" for all "n", then:$int\_S\; liminf\_\{n\; oinfty\}\; f\_n,dmu\; le\; liminf\_\{n\; oinfty\}\; int\_S\; f\_n,dmu,.$

Proof

Apply Fatou's lemma to the non-negative sequence given by "f"_"n" + "g".

Pointwise convergence

If in the previous setting the sequence "f"₁, "f"₂, . . . converges pointwise to a function "f" "μ"-almost everywhere on "S", then:$int\_S\; f,dmu\; le\; liminf\_\{n\; oinfty\}\; int\_S\; f\_n,dmu,.$

Proof

Note that "f" has to agree with the limit inferior of the functions "f"_"n" almost everywhere, and that the values of the integrand on a set of measure zero have no influence on the value of the integral.

Convergence in measure

The last assertion also holds, if the sequence "f"₁, "f"₂, . . . converges in measure to a function "f".

Proof

There exists a subsequence such that:$lim\_\{k\; oinfty\}\; int\_S\; f\_\{n\_k\},dmu=liminf\_\{n\; oinfty\}\; int\_S\; f\_n,dmu,.$Since this subsequence also converges in measure to "f", there exists a further subsequence, which converges pointwise to "f" almost everywhere, hence the previous variation of Fatou's lemma is applicable to this subsubsequence.

Fatou's lemma for conditional expectations

In probability theory, by a change of notation, the above versions of Fatou's lemma are applicable to sequences of random variables "X"₁, "X"₂, . . . defined on a probability space $scriptstyle(Omega,,mathcal\; F,,mathbb\; P)$; the integrals turn into expectations. In addition, there is also a version for conditional expectations.

tandard version

Let "X"₁, "X"₂, . . . be a sequence of non-negative random variables on a probability space $scriptstyle(Omega,mathcal\; F,mathbb\; P)$ and let$scriptstyle\; mathcal\; G,subset,mathcal\; F$ be a sub-σ-algebra. Then:$mathbb\{E\}Bigl\; [liminf\_\{n\; oinfty\}X\_n,Big|,mathcal\; GBigr]\; leliminf\_\{n\; oinfty\},mathbb\{E\}\; [X\_n|mathcal\; G]$ almost surely.

Note: Conditional expectation for non-negative random variables is always well defined, finite expectation is not needed.

Proof

Besides a change of notation, the proof is very similar to the one for the standard version of Fatou's lemma above, however the monotone convergence theorem for conditional expectations has to be applied.

Let "X" denote the limit inferior of the "X"_"n". For every natural number "k" define pointwise the random variable:$Y\_k=inf\_\{nge\; k\}X\_n.$Then the sequence "Y"₁, "Y"₂, . . . is increasing and converges pointwise to "X".For "k" ≤ "n", we have "Y"_"k" ≤ "X"_"n", so that:$mathbb\{E\}\; [Y\_k|mathcal\; G]\; lemathbb\{E\}\; [X\_n|mathcal\; G]$ almost surelyby the monotonicity of conditional expectation, hence:$mathbb\{E\}\; [Y\_k|mathcal\; G]\; leinf\_\{nge\; k\}mathbb\{E\}\; [X\_n|mathcal\; G]$ almost surely,because the countable union of the exceptional sets of probability zero is again a null set.Using the definition of "X", its representation as pointwise limit of the "Y"_"k", the monotone convergence theorem for conditional expectations, the last inequality, and the definition of the limit inferior, it follows that almost surely:

Extension to uniformly integrable negative parts

Let "X"₁, "X"₂, . . . be a sequence of random variables on a probability space $scriptstyle(Omega,mathcal\; F,mathbb\; P)$ and let$scriptstyle\; mathcal\; G,subset,mathcal\; F$ be a sub-σ-algebra. If the negative parts

:$X\_n^-:=max\{-X\_n,0\},qquad\; nin\{mathbb\; N\},$

are uniformly integrable, then

:$mathbb\{E\}Bigl\; [liminf\_\{n\; oinfty\}X\_n,Big|,mathcal\; GBigr]\; leliminf\_\{n\; oinfty\},mathbb\{E\}\; [X\_n|mathcal\; G]$ almost surely.

Note: On the set where

:$X:=liminf\_\{n\; oinfty\}X\_n$

satisfies

:$mathbb\{E\}\; [max\{X,0\},|,mathcal\; G]\; =infty,$

the left-hand side of the inequality is considered to be plus infinity. The conditional expectation of the limit inferior might not be well defined on this set, because the conditional expectation of the negative part might also be plus infinity.

Proof

Let "ε" > 0. Due to uniform integrability, there exists a "c" > 0 such that

:

Since

:$X+cleliminf\_\{n\; oinfty\}(X\_n+c)^+,$

where "x"⁺ := max{"x",0} denotes the positive part of a real "x", monotonicity of conditional expectation (or the above convention) and the standard version of Fatou's lemma for conditional expectations imply

:$mathbb\{E\}\; [X,|,mathcal\; G]\; +clemathbb\{E\}Bigl\; [liminf\_\{n\; oinfty\}(X\_n+c)^+,Big|,mathcal\; GBigr]\; leliminf\_\{n\; oinfty\}mathbb\{E\}\; [(X\_n+c)^+,|,mathcal\; G]$ almost surely.

Since

:$(X\_n+c)^+=(X\_n+c)+(X\_n+c)^-le\; X\_n+c+X\_n^-1\_\{\{X\_n^->c,$

we have

:$mathbb\{E\}\; [(X\_n+c)^+,|,mathcal\; G]\; lemathbb\{E\}\; [X\_n,|,mathcal\; G]\; +c+varepsilon$ almost surely,

hence

:$mathbb\{E\}\; [X,|,mathcal\; G]\; leliminf\_\{n\; oinfty\}mathbb\{E\}\; [X\_n,|,mathcal\; G]\; +varepsilon$ almost surely.

This implies the assertion.

External links

*planetmath reference|id=3678|title=Fatou's lemma

References

* H.L. Royden, "Real Analysis", Prentice Hall, 1988.