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.
Fatou's lemma remains true if its assumptions hold
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.
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
we have By definition of Lebesgue integral and the properties of supremum, 2.
It can be deduced from the definition of Lebesgue integral that if we notice that, for every
First note that the claim holds if f is the indicator function of a set, by monotonicity of measures.
By linearity, this also immediately implies the claim for simple functions.
Recall the closed intervals generate the Borel σ-algebra.
To prove the first claim, write s as a weighted sum of indicator functions of disjoint sets: Then Since the pre-image
, and the monotonicity of Lebesgue integral, we have In accordance with Step 4, as
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 n > x, then fn(x) = 0.
Contrary to Fatou's lemma, this value is strictly less than the integral of the limit (0).
As discussed in § Extensions and variations of Fatou's lemma below, the problem is that there is no uniform integrable bound on the sequence from below, while 0 is the uniform bound from above.
We apply linearity of Lebesgue integral and Fatou's lemma to the sequence
, then Apply Fatou's lemma to the non-negative sequence given by
almost everywhere, hence the previous variation of Fatou's lemma is applicable to this subsubsequence.
In all of the above statements of Fatou's Lemma, the integration was carried out with respect to a single fixed measure
being their pointwise limit inferior, we have Let Then μ(E-K)=0 and Thus, replacing
Define Then An is a nested increasing sequence of sets whose union contains
has finite measure (this is why we needed to consider the two separate cases), Thus, there exists n such that
In probability theory, by a change of notation, the above versions of Fatou's lemma are applicable to sequences of random variables X1, X2, .
be a sequence of non-negative random variables on a probability space
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.
For every natural number k define pointwise the random variable Then the sequence Y1, Y2, .
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 Let X1, X2, .
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 Note: On the set where satisfies the left-hand side of the inequality is considered to be plus infinity.
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 Since we have hence This implies the assertion.