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Let $\{f_n\}$ be a sequence of measurable functions on measurable $E$ that converge pointwise a.e. on $E$ to $f$.

Let $\{g_n\}$ be a sequence of measurable functions on $E$ that converge pointwise a.e. on $E$ to $g$ such that $|f_n| \leq g_n$ for all $n \geq 1$

Knowing that $\lim\limits_{n \rightarrow \infty}$ $\int_E$ $g_n$ = $\int_E$ $g < \infty$, (so $g$ is $a.e.$ finite), how can I build a control integrable

function for all $f_n$ fo be able to apply DCT.

I doubt I can claim that there exists $N>0$ such that $|f_n| \leq g$ for all $n \geq N$, or can I?

ISO
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  • You won't be able to construct such a function. For a proof of the statement, see General Lebesgue Dominated Convergence Theorem – nicoyanovsky Feb 22 '23 at 02:40
  • @nicoyanovsky Thank you. I am familiar with the general proof using Fatou's Lemma. I think that the goal of the exercise is to train on how to develop a control function from a sequence of "measurable" functions. – ISO Feb 22 '23 at 04:59
  • I don't think there's a way to do that for general sequences $f_n$ and $g_n$. If you want an explicit way to construct the dominant function you might need some more information about the $f_n$ and $g_n$ – nicoyanovsky Feb 22 '23 at 05:07
  • @nicoyanovsky that's what I thought. All the info regarding $f_n$ and $g_n$ were included in the question. – ISO Feb 22 '23 at 05:38

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