Ex2-13

**Solution to Chapter 2, Exercise 13**

Theorem 2.16 (i)

If

∫
a
b

f

d
ϕ

{\displaystyle \int\_a^b f~d\phi}

exists, then so do

∫
a
b

c
f

d
ϕ

{\displaystyle \int\_a^b cf~d\phi}

 and

∫
a
b

f

d
(
c
ϕ
)

{\displaystyle \int\_a^b f~d(c\phi)}

 for any constant c, and

∫
a
b

c
f

d
# ϕ

∫
a
b

f

d
(
c
ϕ
# )
c

∫
a
b

f

d
ϕ
.

{\displaystyle \int\_a^b cf~d\phi=\int\_a^b f~d(c\phi)=c\int\_a^b f~d\phi.}

Proof:

Consider a partition

# a

x
0

<

x
1

<
…
<

x
m

=
b
}

{\displaystyle \Gamma=\{a=x\_0<x\_1<\ldots<x\_m=b\}}

of

[
a
,
b
]

{\displaystyle [a, b]}

 and arbitrarily select intermediate points

{

ζ
i

}

# i
1

m

{\displaystyle \{\zeta\_i\}\_{i=1}^m}

, where

x

i
−
1

≤

ζ
i

≤

x
i

{\displaystyle x\_{i-1}\leq \zeta\_i\leq x\_i}

.

R
Γ

(
c
f
)

=

∑

# i
1

m

c
f
(

ζ
i

)
[
ϕ
(

x
i

)
−
ϕ
(

x

i
−
1

)
]

=
c

(

∑

# i
1

m

f
(

ζ
i

)
[
ϕ
(

x
i

)
−
ϕ
(

x

i
−
1

)
]

)

=
c

R
Γ

(
f
)
.

{\displaystyle \begin{align}
R\_\Gamma(cf)&=\sum\_{i=1}^m cf(\zeta\_i)[\phi(x\_i)-\phi(x\_{i-1})]\\
&=c\left(\sum\_{i=1}^m f(\zeta\_i)[\phi(x\_i)-\phi(x\_{i-1})]\right)\\
&=cR\_\Gamma(f).
\end{align}}

Then for any sequence of partitions

{

Γ
k

}

{\displaystyle \{\Gamma\_k\}}

with

lim

k
→
∞

g
a
p
(

Γ
k

# )
0

{\displaystyle \lim\_{k\to\infty} gap(\Gamma\_k)=0}

, we have

lim

k
→
∞

R

Γ
k

(
c
f
)

=

lim

k
→
∞

c

R

Γ
k

(
f
)

=
c

lim

k
→
∞

R

Γ
k

(
f
)

=
c

∫
a
b

f

d
ϕ
.

{\displaystyle \begin{align}
\lim\_{k\to \infty}R\_{\Gamma\_k}(cf)&=\lim\_{k\to \infty}cR\_{\Gamma\_k}(f)\\
&=c\lim\_{k\to \infty}R\_{\Gamma\_k}(f)\\
&=c\int\_a^b f~d\phi.
\end{align}}

Since

∫
a
b

f

d
ϕ

{\displaystyle \int\_a^b f~d\phi}

exists, clearly

c

∫
a
b

f

d
ϕ

{\displaystyle c\int\_a^b f~d\phi}

 also exists for any constant c. Thus,

lim

k
→
∞

R

Γ
k

(
c
f
# )

∫
a
b

c
f

d
ϕ

{\displaystyle \lim\_{k\to \infty}R\_{\Gamma\_k}(cf)=\int\_a^b cf~d\phi}

 exists and

∫
a
b

c
f

d
# ϕ
c

∫
a
b

f

d
ϕ

{\displaystyle \int\_a^b cf~d\phi=c\int\_a^b f~d\phi}

.
Similarly,

R
Γ

(
f
,
c
ϕ
)

=

∑

# i
1

m

f
(

ζ
i

)
[
c
ϕ
(

x
i

)
−
c
ϕ
(

x

i
−
1

)
]

=

∑

# i
1

m

f
(

ζ
i

)
c
[
ϕ
(

x
i

)
−
ϕ
(

x

i
−
1

)
]

=
c

(

∑

# i
1

m

f
(

ζ
i

)
[
ϕ
(

x
i

)
−
ϕ
(

x

i
−
1

)
]

)

=
c

R
Γ

(
f
,
ϕ
)
.

{\displaystyle \begin{align}
R\_\Gamma(f, c\phi)&=\sum\_{i=1}^m f(\zeta\_i)[c\phi(x\_i)-c\phi(x\_{i-1})]\\
&=\sum\_{i=1}^m f(\zeta\_i)c[\phi(x\_i)-\phi(x\_{i-1})]\\
&=c\left(\sum\_{i=1}^m f(\zeta\_i)[\phi(x\_i)-\phi(x\_{i-1})]\right)\\
&=cR\_\Gamma(f,\phi).
\end{align}}

For any sequence of partitions

{

Γ
k

}

{\displaystyle \{\Gamma\_k\}}

with

lim

k
→
∞

g
a
p
(

Γ
k

# )
0

{\displaystyle \lim\_{k\to\infty} gap(\Gamma\_k)=0}

,

lim

k
→
∞

R

Γ
k

(
f
,
c
ϕ
)

=

lim

k
→
∞

c

R

Γ
k

(
f
,

ϕ
)

=
c

lim

k
→
∞

R

Γ
k

(
f
,

ϕ
)

=
c

∫
a
b

f

d
ϕ
.

{\displaystyle \begin{align}
\lim\_{k\to \infty}R\_{\Gamma\_k}(f,c\phi)&=\lim\_{k\to \infty}cR\_{\Gamma\_k}(f,~\phi)\\
&=c\lim\_{k\to \infty}R\_{\Gamma\_k}(f,~\phi)\\
&=c\int\_a^b f~d\phi.
\end{align}}

Since

∫
a
b

f

d
ϕ

{\displaystyle \int\_a^b f~d\phi}

exists, clearly

c

∫
a
b

f

d
ϕ

{\displaystyle c\int\_a^b f~d\phi}

 also exists for any constant c. Thus,

lim

k
→
∞

R

Γ
k

(
f
,
c
ϕ
# )

∫
a
b

f

d
c
ϕ

{\displaystyle \lim\_{k\to \infty}R\_{\Gamma\_k}(f,c\phi)=\int\_a^b f~dc\phi}

 exists and

∫
a
b

f

d
(
c
ϕ
# )
c

∫
a
b

f

d
ϕ

{\displaystyle \int\_a^b f~d(c\phi)=c\int\_a^b f~d\phi}

. This completes the proof.

Proof of Theorem 2.16 (ii)

If

∫
a
b

f
1

d

ϕ

{\displaystyle \int\_a^b f\_1 \operatorname{d}\!\phi}

 and

∫
a
b

f
2

d

ϕ

{\displaystyle \int\_a^b f\_2 \operatorname{d}\!\phi}

 both exist, so does

∫
a
b

(

f
1

f
2

)
d

ϕ

{\displaystyle \int\_a^b (f\_1+f\_2) \operatorname{d}\!\phi}

, and

∫
a
b

(

f
1

f
2

)
d

# ϕ

∫
a
b

f
1

d

ϕ

∫
a
b

f
2

d

ϕ
.

{\displaystyle \int\_a^b (f\_1+f\_2) \operatorname{d}\!\phi = \int\_a^b f\_1 \operatorname{d}\!\phi + \int\_a^b f\_2 \operatorname{d}\!\phi.}

Let

Γ

{\displaystyle \Gamma}

 be a partition

# Γ
{

x
0

,

x
1

,
…
,

x
n

}

{\displaystyle \Gamma=\{x\_0,x\_1,\dots,x\_n\}}

 of

[
a
,
b
]

{\displaystyle [a, b]}

, the sample point

{

ξ
i

}

{\displaystyle \{\xi\_i\}}

 is

ξ
i

∈
[

x

i
−
1

,

x
i

]

{\displaystyle \xi\_i \in [x\_{i-1}, x\_i]}

. Then

R
Γ

(

f
1

f
2

)

=

∑

# i
1

m

[

f
1

(

ξ
1

)

f
2

(

ξ
i

)
]
[
ϕ
(

x
i

)
−
ϕ
(

x

i
−
1

)
]

=

∑

# i
1

m

f
1

(

ξ
1

)
[
ϕ
(

x
i

)
−
ϕ
(

x

i
−
1

)
]

∑

# i
1

m

f
2

(

ξ
i

)
[
ϕ
(

x
i

)
−
ϕ
(

x

i
−
1

)
]

=

R
Γ

(

f
1

)

R
Γ

(

f
2

)
.

{\displaystyle \begin{align}
R\_\Gamma (f\_1+f\_2) &= \sum\_{i=1}^m [f\_1(\xi\_1)+f\_2(\xi\_i)][\phi(x\_i)-\phi(x\_{i-1})] \\
&= \sum\_{i=1}^m f\_1(\xi\_1)[\phi(x\_i)-\phi(x\_{i-1})] + \sum\_{i=1}^m f\_2(\xi\_i)[\phi(x\_i)-\phi(x\_{i-1})] \\
&= R\_\Gamma(f\_1)+R\_\Gamma(f\_2).
\end{align}}

Then for any sequence of partitions

{

Γ
k

}

{\displaystyle \{\Gamma\_k\}}

,

lim

k
→
∞

g
a
p
(

Γ
k

# )
0

{\displaystyle \lim\_{k\to\infty} gap(\Gamma\_k)=0}

,

lim

k
→
∞

R

Γ
k

(

f
1

f
2

)

=

lim

k
→
∞

R

Γ
k

(

f
1

)

lim

k
→
∞

R

Γ
k

(

f
2

)

=

∫
a
b

f
1

d

ϕ

∫
a
b

f
2

d

ϕ
.

{\displaystyle \begin{align}
\lim\_{k\to\infty} R\_{\Gamma\_k}(f\_1+f\_2) &= \lim\_{k\to\infty} R\_{\Gamma\_k}(f\_1) + \lim\_{k\to\infty} R\_{\Gamma\_k}(f\_2) \\
&= \int\_a^b f\_1 \operatorname{d}\!\phi + \int\_a^b f\_2 \operatorname{d}\!\phi.
\end{align}}

Thus

lim

k
→
∞

R

Γ
k

(

f
1

f
2

)

{\displaystyle \lim\_{k\to\infty}R\_{\Gamma\_k}(f\_1+f\_2)}

 exists and so

∫
a
b

(

f
1

f
2

)
d

ϕ

{\displaystyle \int\_a^b (f\_1+f\_2) \operatorname{d}\!\phi}

. Also,

∫
a
b

(

f
1

f
2

)
d

# ϕ

∫
a
b

f
1

d

ϕ

∫
a
b

f
2

d

ϕ
.

{\displaystyle \int\_a^b (f\_1+f\_2) \operatorname{d}\!\phi = \int\_a^b f\_1 \operatorname{d}\!\phi + \int\_a^b f\_2 \operatorname{d}\!\phi.}

Proof of Theorem 2.16 (iii)

We want to prove that if

∫
a
b

f

d

ϕ
1

{\displaystyle \int\_a^b f~d\phi\_1}

and

∫
a
b

f

d

ϕ
2

{\displaystyle \int\_a^b f~d\phi\_2}

 exist, so does

∫
a
b

f

(
d

ϕ
1

d

ϕ
2

)

{\displaystyle \int\_a^b f~(d\phi\_1+d\phi\_2)}

, and

∫
a
b

f

(
d

ϕ
1

d

ϕ
2

# )

∫
a
b

f

d

ϕ
1

∫
a
b

f

d

ϕ
2

.

{\displaystyle
\int\_a^b f~(d\phi\_1+d\phi\_2)=\int\_a^b f~d\phi\_1+\int\_a^b f~d\phi\_2.
}

Suppose

# Γ
{

x
0

,

x
1

,
.
.
.
,

x
m

}

{\displaystyle \Gamma=\{x\_0,x\_1,...,x\_m\}}

is a partition of

[
a
,
b
]

{\displaystyle [a, b]}

and

{

ξ
i

}

# i
1

m

{\displaystyle \{\xi\_i\}\_{i=1}^m}

are intermediate points satisfying

x

i
−
1

≤

ξ
i

≤

x
i

{\displaystyle x\_{i-1}\leq \xi\_i\leq x\_i}

.

R
Γ

(

ϕ
1

ϕ
2

)

=

∑

# i
1

m

f
(

ξ
i

)
[

ϕ
1

(

x
i

)

ϕ
2

(

x
i

)
−

ϕ
1

(

x

i
−
1

)
−

ϕ
2

(

x

i
−
1

)
]

=

∑

# i
1

m

f
(

ξ
i

)
[

ϕ
1

(

x
i

)
−

ϕ
1

(

x

i
−
1

)
]

∑

# i
1

m

f
(

ξ
i

)
[

ϕ
2

(

x
i

)
−

ϕ
2

(

x

i
−
1

)
]

=

R
Γ

(

ϕ
1

)

R
Γ

(

ϕ
2

)

{\displaystyle
\begin{align}
R\_\Gamma(\phi\_1+\phi\_2)&=\sum\_{i=1}^m f(\xi\_i)[\phi\_1(x\_i)+\phi\_2(x\_i)-\phi\_1(x\_{i-1})-\phi\_2(x\_{i-1})] \\
&=\sum\_{i=1}^m f(\xi\_i)[\phi\_1(x\_i)-\phi\_1(x\_{i-1})] + \sum\_{i=1}^m f(\xi\_i)[\phi\_2(x\_i)-\phi\_2(x\_{i-1})]\\
&=R\_\Gamma(\phi\_1)+R\_\Gamma(\phi\_2)
\end{align}
}

Then for any sequence of partitions

{

Γ
k

}

{\displaystyle \{\Gamma\_k\}}

with

lim

k
→
∞

g
a
p
(

Γ
k

# )
0

{\displaystyle \lim\_{k\to\infty} gap(\Gamma\_k)=0}

, we have

lim

k
→
∞

R

Γ
k

(

ϕ
1

ϕ
2

)

=

lim

k
→
∞

R

Γ
k

(

ϕ
1

)

lim

k
→
∞

R

Γ
k

(

ϕ
2

)

=

∫
a
b

f

d

ϕ
1

∫
a
b

f

d

ϕ
2

{\displaystyle
\begin{align}
\lim\_{k\to\infty} R\_{\Gamma\_k}(\phi\_1+\phi\_2)&=\lim\_{k\to\infty}R\_{\Gamma\_k}(\phi\_1) + \lim\_{k\to\infty}R\_{\Gamma\_k}(\phi\_2)\\
&=\int\_a^b f~d\phi\_1+\int\_a^b f~d\phi\_2
\end{align}
}

This shows that

lim

k
→
∞

R

Γ
k

(

ϕ
1

ϕ
2

)

{\displaystyle \lim\_{k\to\infty} R\_{\Gamma\_k}(\phi\_1+\phi\_2)}

exists and hence

∫
a
b

f

(
d

ϕ
1

d

ϕ
2

)

{\displaystyle \int\_a^b f~(d\phi\_1+d\phi\_2)}

exists. Also, we have

∫
a
b

f

(
d

ϕ
1

d

ϕ
2

# )

∫
a
b

f

d

ϕ
1

∫
a
b

f

d

ϕ
2

.

{\displaystyle
\int\_a^b f~(d\phi\_1+d\phi\_2)=\int\_a^b f~d\phi\_1+\int\_a^b f~d\phi\_2.
}