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. .