Search results
Results From The WOW.Com Content Network
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.
The following result is a generalisation of the monotone convergence of non negative sums theorem above to the measure theoretic setting. It is a cornerstone of measure and integration theory with many applications and has Fatou's lemma and the dominated convergence theorem as direct consequence.
Almost sure convergence implies convergence in probability (by Fatou's lemma), and hence implies convergence in distribution. It is the notion of convergence used in the strong law of large numbers. The concept of almost sure convergence does not come from a topology on the space of random variables. This means there is no topology on the space ...
Lebesgue's dominated convergence theorem is a special case of the Fatou–Lebesgue theorem. Below, however, is a direct proof that uses Fatou’s lemma as the essential tool. Since f is the pointwise limit of the sequence (f n) of measurable functions that are dominated by g, it is also measurable and dominated by g, hence it is integrable ...
In mathematics, the Fatou–Lebesgue theorem establishes a chain of inequalities relating the integrals (in the sense of Lebesgue) of the limit inferior and the limit superior of a sequence of functions to the limit inferior and the limit superior of integrals of these functions. The theorem is named after Pierre Fatou and Henri Léon Lebesgue. [1]
Fatou's lemma: If {f k} k ∈ N is a sequence of non-negative measurable functions, then . Again, the value of any of the integrals may be infinite. Dominated convergence theorem : Suppose { f k } k ∈ N is a sequence of complex measurable functions with pointwise limit f , and there is a Lebesgue integrable function g (i.e., g belongs to the ...
Fatou's lemma and the monotone convergence theorem hold if almost everywhere convergence is replaced by (local or global) convergence in measure. If μ is σ-finite, Lebesgue's dominated convergence theorem also holds if almost everywhere convergence is replaced by (local or global) convergence in measure.
Dominated convergence theorem; Vitali convergence theorem; Fichera convergence theorem; Cafiero convergence theorem; Fatou's lemma; Monotone convergence theorem for integrals (Beppo Levi's lemma) Interchange of derivative and integral: Leibniz integral rule