In the real world no one should ever say "Battle Space." It's utterly meaningless. OK, people use the term, but it would seem to signify more ignorance than insight. Terms I use are:

Operational Area . - that defined area of terrain and air space in which an operation will take place. The definition is also time limited.

Area of Responsibility . - that defined area where a unit or sub-unit has responsibility for activity and/or security.

As concerns "owners" is not "ownership" often contested?