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$(f_n)$ is a sequence of integrable functions that converges uniformly to $f$. The question asks to show that if the space has a finite measure, then $lim \int f_n = \int f$. I've tried using the convergence theorems, but it appears to have more to do with using the definition of uniform convegence plus some smart unequality chain. Maybe.

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  • $\begingroup$ I think the Dominated Convergence Theorem here just works where as dominating function you could take $f_N + 1$ for a large enough $N$ such that this is actually an upper bound. $\endgroup$ – Stan Tendijck Feb 21 at 0:44
  • $\begingroup$ Do not use any theorem. Just use definitions as in the answer below. $\endgroup$ – Kavi Rama Murthy Feb 21 at 6:42
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Hint: Assuming your functions are defined on space $X$, argue that $$ \left|\int f_nd\mu-\int fd\mu\right|\le \int|f_n-f|d\mu\le \int \sup_{x\in X}|f_n(x)-f(x)|d\mu $$

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