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.


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

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.