When does $L_1$ convergence imply compact convergence?

Question

Consider a metric space $X$, endow it with its Borel $\sigma$-field $\mathcal{X}$ and a Borel measure $\mu$. Consider measurabe (real-valued) functions $f_n$ and $f$, assume that $$ \Vert f_n - f \Vert_1= \int_X |f_n(x)-f(x)| d\mu(x) \to 0, \quad n \to \infty. $$ Under which conditions the above $L^1$-convergence assumption entails compact convergence, i.e. $$ \sup_{x \in K}|f_n(x)-f(x)|\to 0, \quad n \to \infty, $$ for all compact $K \subset X$? I guess that even continuity conditions on $f_n$ and $f$ may not be sufficient. Am I wrong?