I would like to define the support of a mixed motivic sheaf. This should be something similar to the support of a perverse sheaf.:) Is there any 'classical' definition for the latter?

I suspect that the definition should be something like:

a perverse sheaf $P$ on a space (scheme) $S$ is supported on (or is it better to say 'at'?) a subspace $U$ if for any closed immersion $i:Z\to S$ such that $Z$ is disjoint from $U$ the zeroth (perverse) cohomology of both $Ri^*P$ and $Ri^!P$ is $0$. Does this make sense?

Upd. Possibly the word 'support' is not quite appropriate for me (could you suggest something better?:)), whereas the notion of support should be defined as YBL does below. For the intermediate extension $j_{!*}$ of a perverse sheaf $P$ from an (open) subscheme $U$ of $S$ to $S$ I would like to say that $j_{!*}P$ is supported on $U$. So, I want to say that a constant sheaf on a (connected) smooth variety is supported on any its dense subvariety (or at its generic point).

anyproper closed subvariety $i:Z\rightarrow X$, the complex $i^*K$ is concentrated in perverse degree $<0$ and the complex $i^!K$ is concentrated in perverse degree $>0$. This is a consequence of lemma 4.3.2 of BBD and of the fact that $i^*$ is right $t$-exact for the perverse $t$-structure. (And is related to the fact that $K$ does not determine $U$, as any smaller nonempty open will do.) – Alex Feb 5 '11 at 3:30