Counterexample to an application of Dominated Convergence Theorem

Question

I have proved that if $f_n$ is a sequence of measurable functions over a measure space $(X,\mu)$ and $f_n \to f$ a.e., with $f\in L^1(X)$, then

$$\int |f_n-f|\;d\mu \to 0 \iff \int |f_n|\;d\mu \to \int |f|\;d\mu$$

I am looking for a counterexample where I omit the assumption that $f \in L^1(X)$. I know that the implication that fails should be $\Longleftarrow$ but I can't come up with an example.

Thanks in advance.

Answer 1

Let $X$ be the real line with Lebesgue measure.

Let $f(x) = 1$.

Let $f_n(x) = \chi_{[0,n]}(x)$.

Then $\displaystyle \int |f_n| dx = n \to \infty = \int |f| dx$, but $\displaystyle \int |f_n - f| \, dx = \infty$ for all $n$.