Exercise:
Let $E$ be a measurable set in $\mathbb{R}$ under Lebesgue measure, and let $1<p<\infty$. Suppose $\{f_n\}$ is a bounded sequence in $L^p(E)$ and $f \text{ belongs to } L^p(E) $. Consider the following 4 properties:
(i) $\{f_n\} \rightarrow f$ pointwise a.e. on $E$.
(ii) $\{f_n\} \rightharpoonup f$ in $L^p(E)$.
(iii) $\{\|f_n\|_p\}$ converges to $\{\|f\|_p\}$.
(iv) $\{f_n\} \rightarrow f$ in $L^p(E)$.
If $\{f_n\}$ possesses two of these properties, does a subsequence posses all four properties?
I'm not quite sure where to start. My intuition tells me that that this is false. But I'm not sure as to which properties to start with. Any thoughts would be helpful.
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