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Recall Dominated Convergence Theorem: Let $(f_n)_{n\geq 1}$ be a sequence of measurable functions s.th. $f_n\rightarrow f$ pointwise. Suppose there exists a non-negative integrable function $g$ with $|f_n|\leq g$ for all $n\geq 1$. Then \begin{equation} \lim_{n\rightarrow \infty}\int_Sf_nd\mu=\int_S fd\mu. \end{equation}

My question is that: Is it true $f\leq g$?

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Yes, of course. For all $x$ and $n$, $$f_n(x)\le \lvert f_n(x)\rvert\le g(x)$$ Therefore, for all $x$, $$f(x)=\limsup_{n\to\infty} f_n(x)\le g(x)$$

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