Mutual existence of product integrals in normed rings
Authors: Helton Jon C. 
Published: July 2010

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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue Subscript x Sub Subscript p Subscript Baseline product Overscript x Subscript q Baseline Endscripts left-parenthesis 1 plus upper G right-parenthesis minus product Underscript i equals p plus 1 Overscript q Endscripts left-parenthesis 1 plus upper G Subscript i Baseline right-parenthesis EndAbsoluteValue greater-than epsilon semicolon">

|

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript x right-arrow p Superscript plus Baseline Endscripts upper F left-parenthesis p comma x right-parenthesis comma limit Underscript x right-arrow p Superscript minus Baseline Endscripts upper F left-parenthesis x comma p right-parenthesis comma limit Underscript x right-arrow p Superscript plus Baseline Endscripts upper G left-parenthesis p comma x right-parenthesis">

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].