Is a zero sequence in $L^1$ eventually in $L^p$ for each $p>1$?

Question

Let $\mu$ be a probability measure. I wondered if $(f_n)$ is a sequence in $L^1\equiv L^1(\mu)$ with $f_n\rightarrow 0$ in $L^1$, is this enough for $(f_n)$ to be eventually in $L^q$ for each $q>1$, i.e. that

$$\forall\, q> 1 \ : \ \exists\, n_0 \ \text{ such that } \ (f_n)_{n\geq n_0}\subseteq L^q \ ?$$

Do you know of a quick counterexample?

Answer 1

On $(0,1)$ with Lebesgue measure there exists $f \in L^{1}$ such that $f \notin L^{q}$ for any $ q >1$. Take $f_n=\frac f n$ to get a counter-example.

Explicit example: $f(x)= \sum\limits_{k=2}^{\infty} \frac k {(\ln k)^{2}} I_{(\frac 1 {k+1},\frac 1 k)}$.