Lévy's convergence theorem

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In probability theory Lévy's convergence theorem (sometimes also called Lévy's dominated convergence theorem) states that for the random variables $ξ n$ such that $|\xi_n| < \eta,\; \mathrm{E}\eta < \infty$ and $\xi_n\to\xi\; \mathrm{P}-a.s.$ one has $\mathrm{E}|\xi| < \infty,\; \mathrm{E}\xi_n\to E\xi$ and $\mathrm{E} |\xi-\xi_n|\to 0$ . Essentially, it' the sufficient condition for the almost sure convergence to imply $L 1$ -convergence. The condition $|\xi_n| < \eta,\; \mathrm{E}\eta < \infty$ could be relaxed. Instead, the family ${ξ n}$ should be uniformly integrable.