Let $f \in L^p(\mathbb{R}^n,\mathbb{R}^m)$. If $m=1$ then the essential support of $f$ is a mainstream definition; see here for example. However, when $m>1$ is the following definition used?

$$ \operatorname{ess-supp}(f) := \bigcap \left\{ K \subseteq \mathbb{R}^n : \, K \mbox{ closed and } [f](x)=0 \mbox{ $\mu$-a.e. } x \not\in K \right\}, $$ where here $0$ denotes the zero-vector in $\mathbb{R}^D$?