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Let $\{f_n\} \to f$ pointwise on $E$, then $\int_E f \leq \liminf \int_E f_n$.

The book claims that it suffices to show that if $h$ is any bounded measurable function of finite support for which $0\leq h \leq f$ on E, then $\int_E h \leq \liminf \int_E f_n$. Why is this the case?

Note: this is from page 82 of Royden's real analysis.

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This is enough because, if $A_f$ is the set of bounded measurable functions with finite support and $0\leq h\leq f$, then $$\int_Ef=\sup_{h\in A_f}\int_Eh.$$

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  • $\begingroup$ so if we define $h_n = min\{h, f_n\}$, why does $h_n -> h$? $\endgroup$ – user1559897 May 12 '16 at 2:41

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