If a sequence of functions $f_n$ converges to $f$ in $L^p$, is it necessary that $f$ itself be in $L^p$?

Question

I'm reading Real Analysis for Graduate Students by Richard Bass. One of the exercises is the following:

Prove that if $p$ and $q$ are conjugate exponents, $f_n \to f$ in $L^p(\mu)$ and $g \in L^q(\mu)$, then $$ \int f_n g d\mu \to \int fg d\mu $$

The result follows pretty easily if $f$ is in $L^p(\mu)$ and the $f_n$ are eventually in $L^p$, but is this the case? Is it possible for $| |f_n - f| |_p \to 0$ with $f \not\in L^p(\mu)$? I've searched, but no text I've found addresses this issue.

Answer 1

So for some $n$ we must have $\|f_{n}-f\|_{L^{p}}< 1$. Now we have $\|f\|_{L^{p}}\leq\|f_{n}-f\|_{L^{p}}+\|f_{n}\|_{L^{p}}<1+\|f_{n}\|_{L^{p}}<\infty$.

Answer 2

It is against conventions to say that $f_n \to f$ in $L^{p}$ if $\|f_n-f\|_p \to 0$ without $f_n$ and $f$ belonging to $L^{p}$. In any topological space we talk about convergence in the space only for elements of that space.

If it is assumed that $\{f_n\} \subset L^{q}$ and $(f_n)$ converges in $L^{q}$ the we can say that there exists $f \in L^{q}$ such that $f_n \to f$ in $L^{q}$. But we cannot make the statement without knowing that $f \in L^{q}$