  From mostly a measure-theoretic consideration, we show that for every
nonnegative, finite, and $L^{1}$ function on a given finite measure space there
is some nontrivial sequence of real numbers such that the series, obtained from
summing over the term-by-term products of the reals and the summands of any
divergent series with positive, vanishing summands such as the harmonic series,
is convergent and no greater than the integral of the function. In terms of
inequalities, the implications add additional information on mathematical
expectation and the behavior of divergent series with positive, vanishing
summands, and establish in a broad sense some new, unexpected connections
between probability theory and, for instance, number theory.
