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