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I have a sequence of functions $f_n \to f$. I have shown that both $f_n$ and $f$ are integrable. In order to complete my proof, I needed to apply the dominated convergence theorem to yield

$$\int f_n d\mu \to \int f d\mu$$

Fortunately for me, since I was told $f$ was integrable, and I constructed $f_n$ so that $|f_n|\leq |f|$, I could simply take the dominating function as $f$.

My question however is what if we were not told $f$ was integrable. What if we only knew:

  1. Each $f_n$ itself is integrable

  2. $f_n \to f$

Can you always construct a dominating integrable function then? Clearly if only $(1)$ then holds plenty of counter examples exist. But I am curious if the fact $f_n$ converges to $f$ ensures such a dominating integrable function exists.

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