- #1

joypav

- 151

- 0

**Problem:**

Assume $E$ is a measurable set, $1 \leq p < \infty$, and $f_n \rightarrow f$ in $L^p(E)$. Show that there is a subsequence $(f_{n_k})$ and a function $g \in L^p(E)$ for which $\left| f_{n_k} \right| \leq g$ a.e. on $E$ for all $k$.

**Proof:**

Maybe use?:

- $f_n \rightarrow f$ in $L^p(E)$ iff $lim_{n\rightarrow \infty}\int_{E}\left| f_n \right|^p = \int_{E}\left| f \right|^p $
- In class we have covered the proof for the Riesz-Fischer theorem. So applying that theorem gives the existence of a subsequence $(f_{n_k})$ in $L^p(E)$ so that $f_{n_k} \rightarrow f$ a.e. However, I'm thinking this isn't what I need because we don't necessarily need the subsequence to converge to $f$.

I'm not sure how to show existence of such a $g$... perhaps using Lusin's theorem? I am a bit confused by the relationship between $f$ and $g$. I haven't seen much on approximation in $L^p$ spaces.

Any help, hints, or helpful theorems would be much appreciated!