Notation
We imply
if
.
Proposition
(Vague precompactness of s.p.m.) For
any sequence
of s.p.m. there is a subsequence
that converges vaguely to a s.p.m.
Proof
A function
may be
approximated
by a sequence of piecewise constant function. For the piecewise functions the
statement is apparent. This argument works in both directions.
Definition
(Tight sequence). The sequence of measures
is called "tight"
if
Proposition
(Precompactness of a
tight sequence of p.m.). Let
be a tight sequence of p.m. The there is a subsequence that converges vaguely
to a p.m..
