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][1].  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 $\mathbb{R}^D$?


  [1]: https://en.wikipedia.org/wiki/Support_(mathematics)#Essential_support