 Lq inclusion implies finite measure on a Euclidean Borel subset

Let $X\subseteq {\mathbb{R}}^n$ be Borel. If $L^q(X)\subseteq L^p(X)$ for $1\le p<q\le \infty$, the closed graph theorem makes the inclusion continuous. Applying its norm bound to indicators of $X\cap B(0,R)$ shows that their measures are uniformly bounded, hence $\mu (X)<\infty$.