  We prove a Leibniz rule for BV functions in a complete metric space that is
equipped with a doubling measure and supports a Poincar\'e inequality. Unlike
in previous versions of the rule, we do not assume the functions to be locally
essentially bounded and the end result does not involve a constant $C\ge 1$,
and so our result seems to be essentially the best possible. In order to obtain
the rule in such generality, we first study the weak* convergence of the
variation measure of BV functions, with quasi semicontinuous test functions.
