A sequence of measurable functions that is Cauchy with respect to $\Vert \cdot \Vert_1$?

Question

Suppose that on a measure space $(X,\mathcal{M},\mu)$ I have a sequence of real-valued measurable functions $\{f_n\}_{n\geq 1}$ which is Cauchy with respect to the $L^1$ metric, i.e. $$\forall \ \varepsilon > 0, \ \exists \ N > 0 \ \text{ such that } \ \forall \ m,n > N, \ \ \int_X |f_m(x) - f_n(x)| d\mu(x) < \varepsilon.$$

Can I then conclude that all but finitely many elements of $\{f_n\}$ are integrable, i.e. elements of $L^1$?

If not, does adding the assumption that $\mu$ is $\sigma$-finite help?

Thanks.

Answer 1

No. Consider Lebesgue measure on $(0,1]$ and $f_n(x) =f_0(x)=1/x$ for all $n \in N$ and for all $x\in (0,1]$ . Or if you don't want a constant series, let $f_n(x)=x^{-1}+2^{-n}.$