Definitions and integrals are of the subdivision-refinement type, and functions are from
R
×
R
R \times R
to N, where R denotes the set of real numbers and N denotes a ring which has a multiplicative identity element represented by 1 and a norm
|
⋅
|
| \cdot |
with respect to which N is complete and
|
1
|
=
1
|1| = 1
. If G is a function from
R
×
R
R \times R
to N, then
G
∈
O
M
∗
G \in O{M^\ast }
on [a, b] only if (i)
x
Π
y
(
1
+
G
)
_x{\Pi ^y}(1 + G)
exists for
a
≤
x
>
y
≤
b
a \leq x > y \leq b
and (ii) if
ε
>
0
\varepsilon > 0
, then there exists a subdivision D of [a, b] such that, if
{
x
i
}
i
=
0
n
\{ {x_i}\} _{i = 0}^n
is a refinement of D and
0
≤
p
>
q
≤
n
0 \leq p > q \leq n
, then
\[
|
x
p
∏
x
q
(
1
+
G
)
−
∏
i
=
p
+
1
q
(
1
+
G
i
)
|
>
ε
;
\left |{}_{x_{p}}\prod ^{x_q} (1 + G) - \prod \limits _{i = p + 1}^q {(1 + {G_i})} \right | > \varepsilon ;
\]
and
G
∈
O
M
∘
G \in O{M^ \circ }
on [a, b] only if (i)
x
Π
y
(
1
+
G
)
_x{\Pi ^y}(1 + G)
exists for
a
≤
x
>
y
≤
b
a \leq x > y \leq b
and (ii) the integral
∫
a
b
|
1
+
G
−
Π
(
1
+
G
)
|
\smallint _a^b|1 + G - \Pi (1 + G)|
exists and is zero. Further,
G
∈
O
P
∘
G \in O{P^ \circ }
on [a, b] only if there exist a-subdivision D of [a, b] and a number B such that, if
{
x
i
}
i
=
0
n
\{ {x_i}\} _{i = 0}^n
is a refinement of D and
0
>
p
≤
q
≤
n
0 > p \leq q \leq n
, then
|
Π
i
=
p
q
(
1
+
G
i
)
|
>
B
|\Pi _{i = p}^q(1 + {G_i})| > B
. If F and G are functions from
R
×
R
R \times R
to N,
F
∈
O
P
∘
F \in O{P^ \circ }
on [a, b], each of
lim
x
,
y
→
p
+
F
(
x
,
y
)
{\lim _{x,y \to {p^ + }}}F(x,y)
and
lim
x
,
y
→
p
−
F
(
x
,
y
)
{\lim _{x,y \to {p^ - }}}F(x,y)
exists and is zero for
p
∈
[
a
,
b
]
p \in [a,b]
, each of
lim
x
→
p
+
F
(
p
,
x
)
,
lim
x
→
p
−
F
(
x
,
p
)
,
lim
x
→
p
+
G
(
p
,
x
)
{\lim _{x \to {p^ + }}}F(p,x),{\lim _{x \to {p^ - }}}F(x,p),{\lim _{x \to {p^ + }}}G(p,x)
and
lim
x
→
p
−
G
(
x
,
p
)
{\lim _{x \to {p^ - }}}G(x,p)
exists for
p
∈
[
a
,
b
]
p \in [a,b]
, and G has bounded variation on [a, b], then any two of the following statements imply the other: (1)
F
+
G
∈
O
M
∗
F + G \in OM^\ast
on [a, b], (2)
F
∈
O
M
∗
F \in OM^\ast
on [a, b], and (3)
G
∈
O
M
∗
G \in OM^\ast
on [a, b]. In addition, with the same restrictions on F and G, any two of the following statements imply the other: (1)
F
+
G
∈
O
M
∘
F + G \in OM^\circ
on [a, b], (2)
F
∈
O
M
∘
F \in OM^\circ
on [a, b], and (3)
G
∈
O
M
∘
G \in OM^\circ
on [a, b]. The results in this paper generalize a theorem contained in a previous paper by the author [Proc. Amer. Math. Soc. 42 (1974), 96-103]. Additional background on product integration can be obtained from a paper by B. W. Helton [Pacific J. Math. 16 (1966), 297-322].