All of this is fine for real-valued inputs and outcomes. It's the number of events (coin flips, measurements, etc) that we're restricting to be countable.
No, the number of events
is necessarily also finite or
uncountable.
Indeed, it is a nice exercise
that there are no countably
infinite sigma algebras
(extra credit for a solution!).
It's just can't take uncountably
many events, take their union,
and assume that the result is
also an event.
