Automata over infinite words provide a powerful framework, which we
can use to solve various decision problems.  However, the automatized
reasoning with restricted classes of automata over infinite words is
often simpler and more efficient.  For instance, weak deterministic
B"uchi automata, which recognize the omega-regular languages in the
Borel class F_sigma/\G_delta, can be handled algorithmically almost as
efficient as deterministic automata over finite words.  In this paper,
we show how and when we can determinize automata over infinite words
by the standard powerset construction for finite words. The presented
construction is more efficient than all-purpose constructions for
automata that recognize languages in F_sigma/\G_delta. Further, based
on the powerset construction, we present an improved automata
construction that handles the quantification in the automata-based
approach for FO(R,Z,+,<)$ much more efficiently.