  The problem of testing two simple hypotheses in a general probability space
is considered. For a fixed type-I error probability, the best exponential decay
rate of the type-II error probability is investigated. In regular asymptotic
cases (i.e., when the length of the observation interval grows without limit)
the best decay rate is given by Stein's exponent. In the paper, for a general
probability space, some non-asymptotic lower and upper bounds for the best rate
are derived. These bounds represent pure analytic relations without any
limiting operations. In some natural cases, these bounds also give the
convergence rate for Stein's exponent. Some illustrating examples are also
provided.
