A mapping theorem for logarithmic and integration-by-parts operators

Abstract:

Suppose U is a set, F is a field of subsets of U, ${\mathfrak {p}_{AB}}$ is the set of all bounded real-valued finitely additive functions defined on F, and W is a collection of functions from F into $\exp ({\mathbf {R}})$, closed under multiplication, each element of which has range union bounded and bounded away from 0. Let $\mathcal {P}$ denote the set to which T belongs iff T is a function from W into ${\mathfrak {p}_{AB}}$ such that if each of $\alpha$ and $\beta$ is in W and V is in F, then the following integrals exist and the following “integration-by-parts” equation holds: \[ \int _V \alpha (I)T(\beta )(I) + \int _V {\beta (I)T(\alpha )(I) = T(\alpha \beta )(V).} \] Let $\mathfrak {L}$ denote the set to which S belongs iff S is a function from W into ${\mathfrak {p}_{AB}}$ such that if each of $\alpha$ and $\beta$ is in W, then the integral $\smallint _U {\alpha (I)S(\beta )(I)}$ exists and the following “logarithmic” equation holds: $S(\alpha \beta ) = S(\alpha ) + S(\beta )$. It is shown that $\{ (T,S):T\;{\text {in}}\;\mathcal {P},\;S = \{ (\alpha ,\;\smallint {(1/\alpha )T(\alpha )):\;\alpha \;{\text {in}}\;W\} \} }$ is a one-one mapping from $\mathcal {P}$ onto $\mathfrak {L}$.