Conditions on Lebesgue's Dominated Convergence In Shiryaev's probability, the formulation of the Lebesgue Dominated Convergence Theorem is as it follows:

Let $\eta, \xi, \xi_1 \xi_2 \ldots $ be random variables such that $| \xi_n | \leq \eta$, $E \eta < \infty$ and $\xi_n \to \xi$ almost-surely. Then $E|\xi| < \infty$  and: $$ E \xi_n \to E \xi $$ and $$ E |\xi_n - \xi | \to 0 $$

I know that if I remove the "dominated" from the DCT, the convergence can fail, but I'm thinking how critical the other conditions of the theorem are to ensure convergence. A few questions arised:

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*If I drop the condition that $\xi_n \to \xi$ almost-surely and choose other modes of convergence, will the theorem still hold? Like, how "strong" should the convergence from $\xi_n \to \xi$ be?


*Why state it in separate that $E \xi_n \to E \xi$ AND $E|\xi_n - \xi| \to 0 $? Is there a case where $E \xi_n \to E \xi$, but $E|\xi_n - \xi| \to 0 $ doesn't hold?

Convergence in probability is enough for the conclusion of DCT to hold. Conversely, if $E|\xi_n-\xi| \to 0$ then $\xi_n \to \xi$ in probability.
If $X\sim N(0,1)$ and $X_n=-X$ for all $n$ then $EX_n \to EX$ but $E|X_n-X|$ does not tend to $0$.
