The Dominated Convergence Theorem is as follows:
What if the sequence $\left\{f_n \right\} \notin L^1$? Could someone provide a counterexample as to why the theorem wouldn't hold? Thanks!
2026-05-14 12:34:39.1778762079
counterexample for Dominated Convergence Theorem
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If you require $|f_{n}|\le g, a.e , g\in L^{1},\forall n$, then you automatically showed $f_{n}\in L^{1}$ as well since $\int |f_{n}|\le \int g\le c$. So this cannot happen. And if $f_{n}$ are not Lesbegue integrable, then I think Did's comment is helpful.