We will answer the following question:
[Q.] Let be a sequence of regular Borel measures on a set
, which converges to a measure
on
in weak-star. Is it possible to show that
converges to
in weak-star?
Recall the definition of weak-star convergence first in this context. Since the dual space of is the Borel regular measures on
(see the Appendix C of book [Conway 1990, A course in functional analysis]),
is convergent to
in weak-star, if
[A.] The answer is NO. Let , and
, where
is the dirac measure. Note that
converge to zero measure
in weak-star, but
converge to
. In particular, the limit of total variation of the sequence is not equal to total variation of the limit measure, i.e.
.