Suppose
U
U
is a set,
F
{\mathbf {F}}
is a field of subsets of
U
,
p
A
B
U,{\mathfrak {p}_{AB}}
is the set of all real-valued, bounded finitely additive functions on
F
{\mathbf {F}}
, and for each
ρ
\rho
in
p
A
B
,
A
ρ
{\mathfrak {p}_{AB}},{\mathcal {A}_\rho }
is the set of all elements of
p
A
B
{\mathfrak {p}_{AB}}
absolutely continuous with respect to
ρ
,
p
A
+
\rho ,\mathfrak {p}_A^ +
is the set of all nonnegative-valued elements of
p
A
B
{\mathfrak {p}_{AB}}
, and
p
B
{\mathfrak {p}_B}
is the set of all functions from
F
{\mathbf {F}}
into
exp
(
R
)
\exp ({\mathbf {R}})
with bounded range union. An extension of a previous absolute continuity characterization theorem of the author (Proc. Amer. Math. Soc. 18 (1967), 94-99) is given in the form of a characterization of those subsets
S
S
of
p
A
B
{\mathfrak {p}_{AB}}
having the property that if each of
ξ
\xi
and
μ
\mu
is in
p
A
+
\mathfrak {p}_A^ +
, then
ξ
\xi
is in
A
μ
{\mathcal {A}_\mu }
iff it is true that if
α
\alpha
is in
p
B
,
∫
U
α
(
I
)
μ
(
I
)
{\mathfrak {p}_B},{\smallint _U}\alpha (I)\mu (I)
and
∫
U
α
(
I
)
ξ
(
I
)
{\smallint _U}\alpha (I)\xi (I)
exist and the function
∫
α
μ
\smallint \alpha \mu
is in
S
S
, then
∫
α
ξ
\smallint \alpha \xi
is in
S
S
.