Do there exist $g_n(x)$ such that $\int |g_n| \to 0$ but $\liminf |g_n(x)| \gt 0 \ \forall x \in (0,1)$?

Question

$g_n(x)$ are taken to be Lebesgue integrable over $\mathbb R$ and the integral is a Lebesgue integral.

I believe the answer is no. I tried to go along the lines of:

Let $\liminf |g_n(x)| = e(x) \gt 0$. Fix $x \in (0,1)$. Then eventually after $n \ge N(x)$, $|g_n(x)| \ge e(x) $.

If $\sup N(x) \lt \infty$, then eventually after $n\ge N^*$, $$\int_{\mathbb R} |g_n(x)| dx \ge \int_0^1 |g_n(x)|dx \ge \int_0^1e(x) dx = \epsilon >0$$ But the problem is, $\sup N(x)$ doesn't have to be finite.

Answer 1

Fatou's lemma states that:

\begin{equation*} \liminf_{n \to \infty} \int \mid g_n \mid \geqslant \int \liminf_{n \to \infty}\mid g_n \mid > 0 \end{equation*}

That is, what is mentionned in the title does not exist.