Ex2-16

**Chapter 2 Question 16**

First we show a special case where

f

{\displaystyle f}

 is continuous except for a jump discontinuity at

a

{\displaystyle a}

,. We define

f
~

{\displaystyle \tilde f}

 to be the continuous function that agrees with

f

{\displaystyle f}

 on

(
a
,
b
]

{\displaystyle (a, b]}

 but

f
~

(
a
# )

lim

x
→

a

f
(
x
)

{\displaystyle \tilde{f}(a)=\lim\_{x\rightarrow a^+}f(x)}

so

f
~

{\displaystyle \tilde f}

 is continuous on

[
a
,
b
]

{\displaystyle [a, b]}

. Then we claim that

∫
a
b

f
d
ϕ

{\displaystyle \int\_a^b fd\phi}

 exists and

∫
a
b

f
d
# ϕ

∫
a
b

f
~

d
ϕ

{\displaystyle \int\_a^bfd\phi=\int\_a^b\tilde{f}d\phi}

Let

f
(
x
)

≤
M

{\displaystyle |f(x)|\leq M}

 for all

x
,
ϕ

{\displaystyle x, \phi}

 s continuous at

a

{\displaystyle a}

. Given

ϵ
>
0

{\displaystyle \epsilon > 0}

, choose

δ
>
0

{\displaystyle \delta > 0}

 such that

ϕ
(
x
)
−
ϕ
(
a
)

<
ϵ

 if

x
−
a

<
δ

Then for any partition

Γ

{\displaystyle \Gamma}

 with

Γ

<
δ

{\displaystyle |\Gamma|<\delta}

, choosing the same

ξ
i

{\displaystyle \xi\_i}

's for

f
,

f
~

{\displaystyle f,\tilde f}

 except possibly the one in

[
a
,

x
1

]

{\displaystyle [a,x\_1]}

, we have

S

Γ

(
f
)
−

S

Γ

(

f
~

)

=

(

f
(
ξ
)
−

f
~

(

ξ
′

)

)

(

ϕ
(

x
1

)
−
ϕ
(
a
)

)

<
2
M
ϵ

{\displaystyle \begin{align}

&<2M\epsilon

\end{align}}

By taking

ϵ

{\displaystyle \epsilon}

 arbitrarily small and noting that

lim

Γ

→
0

S

Γ

(

f
~

)

{\displaystyle \lim\_{|\Gamma|\rightarrow0}S\_{\Gamma}(\tilde{f}

)}

 exists, we conclude

lim

Γ

→
0

S

Γ

(
f
)

=

∫
a
b

f
d
# ϕ

∫
a
b

f
~

d
ϕ

{\displaystyle \lim\_{|\Gamma|\rightarrow0}S\_{\Gamma}\left(f\right)=\int\_a^bfd\phi=\int\_a^b\tilde{f}d\phi}

The case where

f

{\displaystyle f}

 has a jump discontinuity at

b

{\displaystyle b}

 or at both of

a

{\displaystyle a}

 and

b

{\displaystyle b}

 is an easy extension of the argument above.

We return to the original problem. We can assume

ϕ

{\displaystyle \phi}

 is increasing on

[
a
,
b
]

{\displaystyle [a, b]}

 to prove the statement. Note that for a general

ϕ

{\displaystyle \phi}

 of bounded variation, we can write it as

# ϕ

ϕ
1

−

ϕ
2

{\displaystyle \phi=\phi\_1-\phi\_2}

, the difference of two increasing functions. Assume

f
(
x
)

≤
M

{\displaystyle |f(x)|\leq M}

for all

x

{\displaystyle x}

, and suppose

f

{\displaystyle f}

 has jump discontinuities at

x
1

,
⋯
,

x
n

{\displaystyle x\_1, \cdots, x\_n}

, then define

f
0

,

f
1

,
⋯
,

f
n

{\displaystyle f\_0,f\_1,\cdots,f\_n}

 on

[
a
,

x
1

]
,
[

x
1

,

x
2

]
,
⋯
,
[

x
n

,
b
]

{\displaystyle [a,x\_1],[x\_1,x\_2],\cdots,[x\_n,b]}

 such that

f
i

(
x
# )
f
(
x
)

 for
x
∈
[

x
i

,

x

i
1

]

{\displaystyle f\_i(x)=f(x)\quad\text{ for }x\in[x\_i,x\_{i+1}]}

where

# a

x
0

,
# b

x

n
1

{\displaystyle a=x\_0,b=x\_{n+1}}

. Now we claim

∫
a
b

f
d
ϕ

{\displaystyle \int\_a^b fd\phi}

 exists and

∫
a
b

f
d

# i
0

n

∫

x
i

x

i
1

f
i

d
ϕ

{\displaystyle \int\_a^bfd\phi=\sum\_{i=0}^n\int\_{x\_i}^{x\_{i+1}}f\_id\phi}

Given

ϵ
>
0

{\displaystyle \epsilon > 0}

, choose

δ
1

>
0

{\displaystyle \delta\_1>0}

 such that for partition

Γ
i

{\displaystyle \Gamma\_i}

 of

[

x
i

,

x

i
1

]

{\displaystyle [x\_i,x\_{i+1}]}

 with

Γ
i

<

δ
1

{\displaystyle |\Gamma\_i|<\delta\_1}

,

S

Γ
i

(

f
i

)
−

∫

x
i

x

i
1

f
i

d
ϕ

<
ϵ

{\displaystyle \left|S\_{\Gamma\_i}(f\_i)-\int\_{x\_i}^{x\_{i+1}}f\_id\phi\right|<\epsilon}

Let

# D
{

x
1

,

x
2

,
⋯
,

x
n

}

{\displaystyle D=\{x\_1,x\_2,\cdots,x\_n\}}

 and let

ϕ

{\displaystyle \phi}

 be continuous at each point of

D

{\displaystyle D}

. Choose

δ
2

>
0

{\displaystyle \delta\_2>0}

 such that for each

# i
1
,
2
,
⋯
,
n

{\displaystyle i = 1,2,\cdots,n}

ϕ
(
x
)
−
ϕ
(
y
)

<
ϵ

 if
x
,
y
∈
(

x
i

−

δ
2

,

x
i

δ
2

)

{\displaystyle |\phi(x)-\phi(y)|<\epsilon\quad\text{ if }x,y\in(x\_i-\delta\_2,x\_i+\delta\_2)}

we also assume

δ
2

{\displaystyle \delta\_2}

 to be small enough such that each

(

x
i

−

δ
2

,

x
i

δ
2

)

{\displaystyle (x\_i-\delta\_2,x\_i+\delta\_2)}

 are disjoint.

Let

# δ
min
(

δ
1

,

δ
2

)

{\displaystyle \delta=\min(\delta\_1,\delta\_2)}

, for partition

Γ

{\displaystyle \Gamma}

 of

[
a
,
b
]

{\displaystyle [a, b]}

 with

Γ

<
δ

{\displaystyle |\Gamma|<\delta}

, we let

Γ
∪
D

{\displaystyle \Gamma\cup D}

 be the refinement with points

{

x
1

,
⋯
,

x
n

}

{\displaystyle \{x\_1,\cdots,x\_n\}}

.

We can choose two sets of

{

ξ
i

}
,
{

ξ
i
′

}

{\displaystyle \{\xi\_i\}, \{\xi\_i'\}}

 in the Riemann-Stieltjes sum such that

ξ
i

=

ξ
i
′

{\displaystyle \xi\_i=\xi\_i'}

 on intervals of

Γ

{\displaystyle \Gamma}

 which don't contain any points from

D

{\displaystyle D}

. Then if we define

# M

sup

x
∈
[
a
,
b
]

f
(
x
)

{\displaystyle M=\sup\_{x\in[a,b]} |f(x)| }

S

Γ
∪
D

(
f
)
−

S

Γ

(
f
)

≤

∑

# i
1

n

(
(

sup

x
∈
[

a
i

,

b
i

]

f
(
x
)

)

ϕ
(

x
i

)
−
ϕ
(

a
i

)

(

sup

x
∈
[

a
i

,

b
i

]

f
(
x
)

)

ϕ
(

b
i

)
−
ϕ
(

x
i

)

(

sup

x
∈
[

a
i

,

b
i

]

f
(
x
)

)

ϕ
(

b
i

)
−
ϕ
(

a
i

)

)

≤
n
⋅
3
M
⋅
ϵ

&\leq n\cdot 3M \cdot\epsilon\end{align}}

where

[

a
i

,

b
i

]

{\displaystyle [a\_i,b\_i]}

is a subinterval in

Γ

{\displaystyle \Gamma}

 containing

x
i

{\displaystyle x\_i }

 for all

# i
1
,
2
,
…
,
n

{\displaystyle i=1, 2, \ldots,n}

.

Let

Γ
0

,

Γ
1

,
⋯
,

Γ
n

{\displaystyle \Gamma\_0, \Gamma\_1, \cdots, \Gamma\_n}

 be partitions respectively of

[

a
0

,

b
0

]
,
[

a
1

,

b
1

]
,
…
,
[

a
n

,

b
n

]
.

{\displaystyle [a\_0,b\_0], [a\_1,b\_1],\ldots, [a\_n,b\_n].}

 The union of those partitions is

Γ
∪
D

{\displaystyle \Gamma\cup D}

, and then

S

Γ
∪
D

(
f
# )

S

Γ
0

(

f
0

)
⋯

S

Γ
n

(

f
n

)

{\displaystyle S\_{\Gamma\cup D}(f)=S\_{\Gamma\_0}(f\_0)+\cdots+S\_{\Gamma\_n}(f\_n)}

implies

S

Γ
∪
D

(
f
)
−

∑

# i
0

n

∫

x
i

x

i
1

f
i

d
ϕ

<
(
n
1
)
ϵ

{\displaystyle \left|S\_{\Gamma \cup D}(f)-\sum\_{i=0}^n\int\_{x\_i}^{x\_{i+1}}f\_id\phi\right|<(n+1)\epsilon}

implies

S

Γ

(
f
)
−

∑

# i
0

n

∫

x
i

x

i
1

f
i

d
ϕ

<
(
3
n
M
n
1
)
ϵ

{\displaystyle \left|S\_{\Gamma}(f)-\sum\_{i=0}^n\int\_{x\_i}^{x\_{i+1}}f\_id\phi\right|<(3nM+n+1)\epsilon}

choose

ϵ

{\displaystyle \epsilon}

 small enough, and we have completed the proof.