1.

Let { } be a collection of open intervals such that

Now, note that , so by subadditivity and the assumption we have:

. I.e.

foreverycollection of open intervals that cover A. Thus, we have that:

inf{ ; } .

But, the term on the left is m*(A)! Which gives us:

which is only possible in m*(A) = 0 (try it out)

Notice that I never mentioned once that A is bounded

EDIT: Monotonicity replaced with countable sub-additivity