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 lemma can be used to prove the Fatou–Lebesgue theorem and Lebesgue's dominated convergence theorem.
Standard statement In what follows, denotes the -algebra of Borel sets on .
Fatou's lemma remains true if its assumptions hold -almost everywhere. In other words, it is enough that there is a null set such that the values are non-negative for every To see this, note that the integrals appearing in Fatou's lemma are unchanged if we change each function on .
Proof Fatou's lemma does not require the monotone convergence theorem, but the latter can be used to provide a quick and natural proof. A proof directly from the definitions of integrals is given further below.
Via the Monotone Convergence Theorem
let . Then: the sequence is pointwise non-decreasing at any and , . Since , and infima and suprema of measurable functions are measurable we see that is measurable.
By the Monotone Convergence Theorem and property (1), the sup and integral may be interchanged:
where the last step used property (2).
From "first principles" To demonstrate that the monotone convergence theorem is not "hidden", the proof below does not use any properties of Lebesgue integral except those established here and the fact that the functions and are measurable.
Denote by the set of simple -measurable functions such that on .
Now we turn to the main theorem
The proof is complete.
Examples for strict inequality Equip the space with the Borel σ-algebra and the Lebesgue measure. Example for a probability space: Let denote the unit interval. For every natural number define
Example with uniform convergence: Let denote the set of all real numbers. Define
These sequences converge on pointwise (respectively uniformly) to the zero function (with zero integral), but every has integral one.
The role of non-negativity A suitable assumption concerning the negative parts of the sequence f1, f2, . . . 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 and the limit, 0, is reached in a finite number of steps: for every x ≥ 0, if , then fn(x) = 0. However, every function fn has integral −1. Contrary to Fatou's lemma, this value is strictly less than the integral of the limit (0).
As discussed in below, the problem is that there is no uniform integrable bound on the sequence from below, while 0 is the uniform bound from above.
Reverse Fatou lemma Let f1, f2, . . . 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 fn ≤ g for all n, then
Note: Here g integrable means that g is measurable and that .
Sketch of proof We apply linearity of Lebesgue integral and Fatou's lemma to the sequence Since this sequence is defined -almost everywhere and non-negative.
Extensions and variations of Fatou's lemma
Integrable lower bound Let be a sequence of extended real-valued measurable functions defined on a measure space . If there exists an integrable function on such that for all , then
Proof Apply Fatou's lemma to the non-negative sequence given by .
Pointwise convergence If in the previous setting the sequence converges pointwise to a function -almost everywhere on , then
Proof Note that has to agree with the limit inferior of the functions 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 converges in measure to a function .
Proof There exists a subsequence such that
Since this subsequence also converges in measure to , there exists a further subsequence, which converges pointwise to almost everywhere, hence the previous variation of Fatou's lemma is applicable to this subsubsequence.
Fatou's Lemma with Converging Measures Measures with setwise convergence
In all of the above statements of Fatou's Lemma, the integration was carried out with respect to a single fixed measure . Suppose that is a sequence of measures on the measurable space such that (see Convergence of measures) . Then, with non-negative integrable functions and being their pointwise limit inferior, we have
{| class="toccolours collapsible collapsed" width="90%" style="text-align:left" !Proof |- |We will prove something a bit stronger here. Namely, we will allow to converge -almost everywhere on a subset of . We seek to show that
Let . Then μ(E-K)=0 and
Thus, replacing by we may assume that converge to pointwise on . Next, note that for any simple function we have
Hence, by the definition of the Lebesgue Integral, it is enough to show that if is any non-negative simple function less than or equal to , then
Let a be the minimum non-negative value of φ. Define
We first consider the case when . We must have that is infinite since
where M is the (necessarily finite) maximum value of that attains.
Next, we define
We have that
But is a nested increasing sequence of functions and hence, by the continuity from below , . Thus, . At the same time,
proving the claim in this case.
The remaining case is when . We must have that is finite. Denote, as above, by the maximum value of and fix Define
Then An is a nested increasing sequence of sets whose union contains . Thus, is a decreasing sequence of sets with empty intersection. Since has finite measure (this is why we needed to consider the two separate cases),
Thus, there exists n such that
Therefore, since
there exists N such that
Hence, for
At the same time,
Hence,
Combining these inequalities gives that
Hence, sending to 0 and taking the liminf in , we get that
completing the proof. |} Asymptotically uniform integrable functions
The following results use the notion asymptotically uniform integrable (a.u.i). A sequence of measurable -valued functions is a.u.i with respect to a sequence of measures if
Weakly converging measures
A sequence of measures on a metric space converges weakly to a finite measure on M if, for each bounded continuous function on ,
Measures with convergence in total variation
A sequence of finite measures on a measurable space converges in total variation to a measure on if
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 X1, X2, . . . defined on a probability space ; the integrals turn into expectations. In addition, there is also a version for conditional expectations.
Standard version Let X1, X2, . . . be a sequence of non-negative random variables on a probability space and let be a sub-σ-algebra. Then 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 Xn. For every natural number k define pointwise the random variable
Then the sequence Y1, Y2, . . . is increasing and converges pointwise to X. For k ≤ n, we have Yk ≤ Xn, so that almost surely by the monotonicity of conditional expectation, hence 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 Yk, 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 X1, X2, . . . be a sequence of random variables on a probability space and let be a sub-σ-algebra. If the negative parts
are uniformly integrable with respect to the conditional expectation, in the sense that, for ε > 0 there exists a c > 0 such that
, then
almost surely.
Note: On the set where
satisfies
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 with respect to the conditional expectation, there exists a c > 0 such that
Since
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
almost surely.
Since
we have
almost surely,
hence
almost surely.
This implies the assertion.
References