Suppose $f_1(x),f_2(x),\ldots:[0,1]\rightarrow\mathbb{R}$ are measurable functions such that $f_1(x)\geq f_2(x)\geq\ldots$. (infinite sequence) and $\lim_{n\rightarrow\infty}f_n(x)=0$. Is it true that $\lim_{n\rightarrow \infty}\int_0^1 f_n(x)dx=0$?
I wanted to apply the dominated convergence theorem but cannot, because $f_i(x)$ is not necessarily integrable. So is there a counterexample for this?
$$ f_n(x) = \begin{cases} \dfrac{\chi_{[0, 1/n]}}{x} &: x \in (0, 1] \\ 0 &: x = 0\end{cases} $$