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].