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Necessity of Lebegue's convergence criterion

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Necessity of Lebegue's convergence criterion

Hello everyone,

the well-known Lebesgue’s dominated convergence theorem states that pointwise convergence of a sequence of functions implies convergence of the sequence of integrals if an integrable function dominating the sequence of functions almost everywhere can be found. My question: is the existence of such a dominating function also a necessary condition for the convergence of integrals? Or can one think of an example where the sequence of integrals does converge to the expected limit, but no dominating function can be found for the sequence of functions?

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