  We introduce a linear space of finitely additive measures to treat the
problem of optimal expected utility from consumption under a stochastic clock
and an unbounded random endowment process. In this way we establish existence
and uniqueness for a large class of utility maximization problems including the
classical ones of terminal wealth or consumption, as well as the problems
depending on a random time-horizon or multiple consumption instances. As an
example we treat explicitly the problem of maximizing the logarithmic utility
of a consumption stream, where the local time of an Ornstein-Uhlenbeck process
acts as a stochastic clock.
