= Solution
Consider the inclusion map
$$
J:(X,\|\cdot\|_X)\longrightarrow
(\mathcal L^1(\mathbb R^n),\|\cdot\|_1).
$$
Its graph is closed. Indeed, suppose $x_k\to x$ in $X$ and $Jx_k\to y$ in $L^1$. By hypothesis, some subsequence of $(x_k)$ converges almost everywhere to $x$. From its $L^1$ convergence, that subsequence has a further subsequence converging almost everywhere to $y$. Uniqueness of almost-everywhere limits gives $x=y$ as $L^1$ classes.
Both spaces are Banach, so the <closed graph theorem> makes $J$ bounded. Therefore some $C>0$ satisfies
$$
\boxed{\|x\|_1\leq C\|x\|_X\qquad(x\in X)}.
$$
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