- Fatou's lemma
In
mathematics , Fatou's lemma establishes aninequality relating theintegral (in the sense of Lebesgue) of the limit inferior of asequence of functions to the limit inferior of integrals of these functions. The lemma is named after the Frenchmathematician Pierre Fatou (1878 - 1929).Fatou's lemma can be used to prove the
Fatou–Lebesgue theorem and Lebesgue'sdominated 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 aprobability space : Let S= [0,1] denote theunit interval . For everynatural number n define:f_n(x)=egin{cases}n& ext{for }xin (0,1/n),\0& ext{otherwise.}end{cases}
* Example withuniform convergence : Let S denote the set of allreal number s. Define :f_n(x)=egin{cases}frac1n& ext{for }xin [0,n] ,\0& ext{otherwise.}end{cases}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:f_n(x)=egin{cases}-frac1n& ext{for }xin [n,2n] ,\0& ext{otherwise.}end{cases}This sequence converges uniformly on "S" to the zero function (with zero integral) and for every "x" ≥ 0 we even have "fn"("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 "fn" 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 extstyleint_S g,dmu
. 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 ofrandom variables "X"1, "X"2, . . . defined on aprobability space scriptstyle(Omega,,mathcal F,,mathbb P); the integrals turn into expectations. In addition, there is also a version forconditional expectation s.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 letscriptstyle 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:egin{align}mathbb{E}Bigl [liminf_{n oinfty}X_n,Big|,mathcal GBigr] &=mathbb{E} [X|mathcal G] =mathbb{E}Bigl [lim_{k oinfty}Y_k,Big|,mathcal GBigr] =lim_{k oinfty}mathbb{E} [Y_k|mathcal G] \&lelim_{k oinfty} inf_{nge k}mathbb{E} [X_n|mathcal G] =liminf_{n oinfty},mathbb{E} [X_n|mathcal G] .end{align}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 letscriptstyle 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
:mathbb{E}igl [X_n^-1_{{X_n^->cigr]
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.
Wikimedia Foundation. 2010.