As reported in https://en.wikipedia.org/wiki/Support_(mathematics), for a function $f:X \rightarrow \mathbb{R}$ we can define some notion of $supp(f)$, in particular:
If $X$ is only a set, we define the theoretical support,
If $(X, \tau)$ is also equipped with a topological structure, we define the closed support and compact support,
If $(X, \tau, \mu, \mathscr{B})$ has a topological structure and also is equipped by a measure, we define the essential support,
I have two different question:
(1) Why, if it is present an a topological structure, are we interested to define the support of a function as a closed subset of $X$?
(2)The essential support is, in some sense, the natural extencion of the closed support that can work with the function defined $\mu$-almost everywhere. However, we can easily show that:
Thm: If $(X, \tau, \mu, \mathscr{B})$ is second-countably space, Then a measurable non zero function $f:(X, \tau, \mu, \mathscr{B}) \rightarrow \mathbb{R}$ is $\mu$-a.e. zero on $(ess\, supp(f))^C$
Then for a non second-countably space, we are not guarantee of this result. Then is it still a well-formulated notion?