Exercise 4.16 in Brezis' Functional Analysis (Counterexample)

Question

The exercise 4.16 in the Brézis book - Functional Analysis, Sobolev Spaces and PDE's, is as follows:

Let $1<p<\infty$. Let $(f_n)$ be a sequence in $L^p(\Omega)$ such that

(i) $f_n$ is bounded in $L^p(\Omega)$. (ii) $f_n \rightarrow f$ a.e. on $\Omega$.

1. Prove that $f_n \rightharpoonup f$ weakly $\sigma(L^p,L^{p'})$;
2. Same conclusion if assumption (ii) is raplaced by

(ii') $\|f_n - f\|_1\rightarrow 0$.

3. Assume now (i), (ii), and $|\Omega|<\infty$. Prove that $\|f_n -f\|_{q}\rightarrow 0$ for every $q$ with $1\leq q<p$.

My question: Is it possible to build a sequence satisfying (i), (ii), and $|\Omega|<\infty$ and $\|f_n -f\|_q \not\rightarrow 0$ for $q=p$, i.e. $\|f_n -f\|_p \not\rightarrow 0$?

Answer 1

Yes. On $(0,1)$ with Lebesgue measure take $f_n(x)=\sqrt n$ for $x<\frac 1 n$ and $0$ for $x \geq \frac 1 n$. Take $f=0$ and $p=2$.