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 FatouâÂÂLebesgue theorem and Lebesgue's dominated convergence theorem.
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 .
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.
let . Then:
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).
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.
Equip the space with the Borel σ-algebra and the Lebesgue measure.
These sequences converge on pointwise (respectively uniformly) to the zero function (with zero integral), but every has integral one.
A suitable assumption concerning the negative parts of the sequence f<sub>1</sub>, f<sub>2</sub>, . . . of functions is necessary for Fatou's lemma, as the following example shows. Let S denote the half line [0,âÂÂ) with the Borel ÃÂ-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 âÂÂ¥ 0, if , then f<sub>n</sub>(x) = 0. However, every function f<sub>n</sub> 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.
Let f<sub>1</sub>, f<sub>2</sub>, . . . be a sequence of extended real-valued measurable functions defined on a measure space (S,ã,ü). If there exists a non-negative integrable function g on S such that f<sub>n</sub> ⤠g for all n, then
Note: Here g integrable means that g is measurable and that .
We apply linearity of Lebesgue integral and Fatou's lemma to the sequence Since this sequence is defined -almost everywhere and non-negative.
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
Apply Fatou's lemma to the non-negative sequence given by .
If in the previous setting the sequence converges pointwise to a function -almost everywhere on , then
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.
The last assertion also holds, if the sequence converges in measure to a function .
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.
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
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
In probability theory, by a change of notation, the above versions of Fatou's lemma are applicable to sequences of random variables X<sub>1</sub>, X<sub>2</sub>, . . . defined on a probability space ; the integrals turn into expectations. In addition, there is also a version for conditional expectations.
Let X<sub>1</sub>, X<sub>2</sub>, . . . be a sequence of non-negative random variables on a probability space and let be a sub-ÃÂ-algebra. Then
Note: Conditional expectation for non-negative random variables is always well defined, finite expectation is not needed.
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<sub>n</sub>. For every natural number k define pointwise the random variable
Then the sequence Y<sub>1</sub>, Y<sub>2</sub>, . . . is increasing and converges pointwise to X. For k ⤠n, we have Y<sub>k</sub> ⤠X<sub>n</sub>, so that
by the monotonicity of conditional expectation, hence
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<Sub>k</sub>, the monotone convergence theorem for conditional expectations, the last inequality, and the definition of the limit inferior, it follows that almost surely
Let X<sub>1</sub>, X<sub>2</sub>, . . . be a sequence of random variables on a probability space and let be a sub-ÃÂ-algebra. If the negative parts
are uniformly integrable with respect to the conditional expectation, in the sense that, for õ > 0 there exists a c > 0 such that
then
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.
Let õ > 0. Due to uniform integrability with respect to the conditional expectation, there exists a c > 0 such that
Since
where x<sup>+</sup> := 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
Since
we have
hence
This implies the assertion.