Math Problems and solutions
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19.6.3.1 Convergence (in measure, almost everywhere)
3.81 $
19.6.3.2 Convergence (in measure, almost everywhere)
5.08 $
19.6.3.3 Convergence (in measure, almost everywhere)
![Problem: Let \( \left(X, A, \mu\right. \) ) be a measurable space, \( A_{n} \in A, n \in \mathbb{N} \) is the sequence of such measurable sets that \[ \sum_{n=1}^{\infty} \mu\left(A_{n}\right)<\infty . \] Prove that \( \lim \sup _{n \rightarrow \infty} A_{n} \) is measurable and its measure is equal to 0 .](/storage/uploads/task_mls/1084/en/19.6.3.1.png){kind=link}
![Problem: Bring an example of a sequence of such functions \( f_{n} \), satisfying the conditions of Fatou's lemma, that \[ \lim _{n \rightarrow \infty} \int_{A} f_{n}(x) \mu(d x): \] a) doesn't exist, b) exists and isn't equal to \( \int_{A} \lim _{n \rightarrow \infty} f_{n}(x) \mu(d x) \).](/storage/uploads/task_mls/1086/en/19.6.3.3.png){kind=link}