3)
Define $ f(x)=\sum_{n=1}^\infty n\chi_{[2^{-n}, 2^{-(n-1)})}(x) $
Then $ \forall p \in (0, \infty), \ \int_0^1 |f|^p=\sum_{n=1}^\infty \frac{n^p}{2^n}<\infty $ by the ratio test.
But $ ess.sup(f)>n \ \forall \ n \Rightarrow f \not\in L^\infty (0,1) $.
--Wardbc 13:19, 5 July 2008 (EDT)
