Convergence in $L_infty$ and $L_1$ even if infinite measure space

Let $\{f_n\}_{n\in \mathbb{N}}$ be a sequence of measurable functions on a measure space and $f$ measurable.
In the literature, assuming the measure space $X$ has finite measure, if $f_n$ converges to $f$ in $L^{\infty}$-norm , then $f_n$ converges to $f$ in $L^{1}$-norm.

Even if $X$ has infinite measure, does it converge to $f$ in $L^{1}$-norm?

Answer

No. Try $f_n$ the constant function such that $f_n(x)=\frac1n$ for every $x$ in $X$.

For an example where each $g_n$ is in $L^1\cap L^\infty$, $g_n\to0$ in $L^\infty$ and not in $L^1$, try $g_n=\frac1n\mathbf 1_{[0,n^2]}$ on $X=\mathbb R$ with Lebesgue measure.