The notion of integration is one of fundamental importance in advanced analysis. The idea of integration is expanded so as to be applicable to sets more general than subsets of
R
{\displaystyle \mathbb {R} }
Measure Theory
Riemann-Stieltjes integral
edit
The Riemann-Stieltjes integral (or Stieltjes integral ) can be seen as an extention of the idea behind the Darboux integral
Let
f
:
[
a
,
b
]
→
R
{\displaystyle f:[a,b]\to \mathbb {R} }
Let
α
:
[
a
,
b
]
→
R
{\displaystyle \alpha :[a,b]\to \mathbb {R} }
α
(
x
)
{\displaystyle \alpha (x)}
[
a
,
b
]
{\displaystyle [a,b]}
Let
P
{\displaystyle {\mathcal {P}}}
[
a
,
b
]
{\displaystyle [a,b]}
x
k
,
x
k
−
1
∈
P
{\displaystyle x_{k},x_{k-1}\in {\mathcal {P}}}
The Upper Sum of
f
{\displaystyle f}
P
{\displaystyle {\mathcal {P}}}
α
{\displaystyle \alpha }
U
(
f
,
P
,
α
)
=
∑
k
=
1
n
M
k
(
α
(
x
k
)
−
α
(
x
k
−
1
)
)
{\displaystyle U(f,{\mathcal {P}},\alpha )=\sum _{k=1}^{n}M_{k}{\big (}\alpha (x_{k})-\alpha (x_{k-1}){\big )}}
where
M
k
{\displaystyle M_{k}}
The Lower Sum of
f
{\displaystyle f}
P
{\displaystyle {\mathcal {P}}}
α
{\displaystyle \alpha }
L
(
f
,
P
,
α
)
=
∑
k
=
1
n
m
k
(
α
(
x
k
)
−
α
(
x
k
−
1
)
)
{\displaystyle L(f,{\mathcal {P}},\alpha )=\sum _{k=1}^{n}m_{k}{\big (}\alpha (x_{k})-\alpha (x_{k-1}){\big )}}
where
m
k
{\displaystyle m_{k}}
Let
f
:
[
a
,
b
]
→
R
{\displaystyle f:[a,b]\to \mathbb {R} }
Let
α
:
[
a
,
b
]
→
R
{\displaystyle \alpha :[a,b]\to \mathbb {R} }
α
(
x
)
{\displaystyle \alpha (x)}
[
a
,
b
]
{\displaystyle [a,b]}
We say that
f
{\displaystyle f}
Riemann-Stieltjes integrable on
[
a
,
b
]
{\displaystyle [a,b]}
α
{\displaystyle \alpha }
sup
P
{
L
(
f
,
P
,
α
)
}
=
inf
P
{
U
(
f
,
P
,
α
)
}
{\displaystyle \sup _{\mathcal {P}}{\Big \{}L(f,{\mathcal {P}},\alpha ){\Big \}}=\inf _{\mathcal {P}}{\Big \{}U(f,{\mathcal {P}},\alpha ){\Big \}}}
where the supremum and the infimum have been taken over the set of all partitions.
L
=
sup
P
{
L
(
f
,
P
,
α
)
}
=
inf
P
{
U
(
f
,
P
,
α
)
}
{\displaystyle L=\sup _{\mathcal {P}}{\Big \{}L(f,{\mathcal {P}},\alpha ){\Big \}}=\inf _{\mathcal {P}}{\Big \{}U(f,{\mathcal {P}},\alpha ){\Big \}}}
f
{\displaystyle f}
[
a
,
b
]
{\displaystyle [a,b]}
α
{\displaystyle \alpha }
∫
a
b
f
(
x
)
d
α
(
x
)
{\displaystyle \int \limits _{a}^{b}f(x)d\alpha (x)}
∫
a
b
f
d
α
{\displaystyle \int \limits _{a}^{b}fd\alpha }
Observe that putting
α
(
x
)
=
x
{\displaystyle \alpha (x)=x}
Henstock Kurtzweil integral
edit
While calculating the Riemann integral , the "fineness" of a partition was measured by it norm . However, it turns out that the norm is a very crude measure for a partition. Thus, by introducing the clever notion of gauges , we can extend the idea of the Riemann integral to a larger class of functions. In fact, it turns out that this integral, called the Henstock-Kurtzweil integral (after Ralph Henstock and Jaroslav Kurzweil ) or Generalised Riemann integral is more general than the Riemann-Stieltjes integral and several other integrals on real intervals.
A Gauge is said to be a function
δ
:
[
a
,
b
]
→
R
+
{\displaystyle \delta :[a,b]\to \mathbb {R} ^{+}}
δ
(
x
)
{\displaystyle \delta (x)}
A tagged partition
P
˙
=
{
(
t
k
,
[
x
k
−
1
,
x
k
]
)
}
k
=
1
n
{\displaystyle {\mathcal {\dot {P}}}={\Big \{}(t_{k},[x_{k-1},x_{k}]){\Big \}}_{k=1}^{n}}
δ-fine for a gauge
δ
{\displaystyle \delta }
k
{\displaystyle k}
[
x
k
−
1
,
x
k
]
⊆
(
t
k
−
δ
(
t
k
)
,
t
k
+
δ
(
t
k
)
)
{\displaystyle [x_{k-1},x_{k}]\subseteq {\big (}t_{k}-\delta (t_{k}),t_{k}+\delta (t_{k}){\big )}}
Let
f
:
[
a
,
b
]
→
R
{\displaystyle f:[a,b]\to \mathbb {R} }
Let
L
∈
R
{\displaystyle L\in \mathbb {R} }
Then,
f
{\displaystyle f}
Henstock-Kurtzweil integrable on
[
a
,
b
]
{\displaystyle [a,b]}
ε
>
0
{\displaystyle \varepsilon >0}
δ
:
[
a
,
b
]
→
R
+
{\displaystyle \delta :[a,b]\to \mathbb {R} ^{+}}
P
˙
{\displaystyle {\mathcal {\dot {P}}}}
[
a
,
b
]
{\displaystyle [a,b]}
|
S
(
f
,
P
˙
)
−
L
|
<
ε
{\displaystyle {\Big |}S(f,{\mathcal {\dot {P}}})-L{\Big |}<\varepsilon }
![{\displaystyle f:[a,b]\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b592d102ccd1ba134d401c5b3ea177baaba3ffac)
![{\displaystyle \alpha :[a,b]\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/37c39ab9d12a69c4a49af6fb4a0fb6eb36d01ff0)
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\displaystyle \delta :[a,b]\to \mathbb {R} ^{+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a938b577a4ffd21873d9c053613e374bf4c8afd)
![{\displaystyle {\mathcal {\dot {P}}}={\Big \{}(t_{k},[x_{k-1},x_{k}]){\Big \}}_{k=1}^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e41d184fe07563ef4008e95e6c9b22efc8d208ea)
![{\displaystyle [x_{k-1},x_{k}]\subseteq {\big (}t_{k}-\delta (t_{k}),t_{k}+\delta (t_{k}){\big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b49a653383e1e166de2b48d14846c5db71ce7f6)
