edit
Let X , Y be measure spaces with measures μ , ν respectively.
Let
f
:
X
×
Y
→
R
∪
{
+
∞
,
−
∞
}
{\displaystyle f:X\times Y\to \mathbb {R} \cup \left\{+\infty ,-\infty \right\}}
Then it is true that
∫
X
(
∫
Y
f
(
x
,
y
)
d
ν
(
y
)
)
d
μ
(
x
)
=
∫
Y
(
∫
X
f
(
x
,
y
)
d
μ
(
x
)
)
d
ν
(
y
)
=
∫
X
×
Y
f
(
x
,
y
)
d
μ
×
ν
(
x
,
y
)
{\displaystyle \int _{X}\left(\int _{Y}f(x,y)\,\mathrm {d} \nu (y)\right)\,\mathrm {d} \mu (x)=\int _{Y}\left(\int _{X}f(x,y)\,\mathrm {d} \mu (x)\right)\,\mathrm {d} \nu (y)=\int _{X\times Y}f(x,y)\,\mathrm {d} \mu \times \nu (x,y)}
provided one of the following criteria:
(Fubini's theorem ) The spaces X , Y are complete (all null sets are measurable), and
f
∈
L
1
(
μ
×
ν
)
{\displaystyle f\in L^{1}\left(\mu \times \nu \right)}
(Tonelli's theorem ) The spaces X , Y are σ-finite (a countable union of finite-measure sets)*, and f ≥ 0.(*) For probability spaces this is automatic.
Convergence of integrals
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Let Ω be a measure space with a measure μ .
Let fn : Ω → ℝ be a sequence of measurable functions that converges pointwise (everywhere, or μ -almost everywhere if μ is a complete measure ) to a function f : Ω → ℝ.
Then it is true that
∫
Ω
f
n
d
μ
→
∫
Ω
f
d
μ
{\displaystyle \int _{\Omega }f_{n}\,d\mu \to \int _{\Omega }f\,d\mu }
(Monotone convergence theorem )
0
≤
f
1
≤
f
2
≤
…
{\displaystyle 0\leq f_{1}\leq f_{2}\leq \ldots }
μ -almost everywhere in Ω.
Note: If additionally
f
∈
L
1
(
μ
)
{\displaystyle f\in L^{1}(\mu )}
f
n
→
f
{\displaystyle f_{n}\to f}
L 1 (μ ) by Scheffé’s lemma .
(Dominated convergence theorem )
|
f
n
|
≤
g
{\displaystyle \left|f_{n}\right|\leq g}
g
∈
L
1
(
μ
)
{\displaystyle g\in L^{1}\left(\mu \right)}
μ -almost everywhere if μ is a complete measure ).
Note: This also gives us
f
n
→
f
{\displaystyle f_{n}\to f}
L 1 (μ ), and
|
|
f
n
|
|
L
1
(
μ
)
↑
|
|
f
|
|
L
1
(
μ
)
≤
|
|
g
|
|
L
1
(
μ
)
{\displaystyle ||f_{n}||_{L^{1}(\mu )}\uparrow ||f||_{L^{1}(\mu )}\leq ||g||_{L^{1}(\mu )}}
(Bounded convergence theorem )
μ
(
Ω
)
<
∞
{\displaystyle \mu (\Omega )<\infty }
|
f
n
|
≤
M
{\displaystyle \left|f_{n}\right|\leq M}
Note: This also gives us
f
n
→
f
{\displaystyle f_{n}\to f}
L 1 (μ ), and
|
|
f
n
|
|
L
1
(
μ
)
↑
|
|
f
|
|
L
1
(
μ
)
≤
M
μ
(
Ω
)
{\displaystyle ||f_{n}||_{L^{1}(\mu )}\uparrow ||f||_{L^{1}(\mu )}\leq M\mu \left(\Omega \right)}
Corollary: Differentiation under the integral sign
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Let
F
(
x
)
:=
∫
Ω
f
(
x
,
ω
)
d
μ
(
ω
)
{\displaystyle F(x):=\int _{\Omega }f(x,\omega )\,d\mu (\omega )}
x
∈
R
{\displaystyle x\in \mathbb {R} }
ω is held constant, for all ω (or μ -almost all ω if μ is a complete measure ), f is differentiable in x . Suppose F is defined in a neighborhood of 0.
Then it is true that
F
′
(
0
)
=
∫
Ω
∂
f
∂
x
(
0
,
ω
)
d
μ
(
ω
)
{\displaystyle F'(0)=\int _{\Omega }{\frac {\partial f}{\partial x}}\left(0,\omega \right)\,d\mu (\omega )}
∂
f
∂
x
(
0
,
ω
)
∈
L
1
(
μ
)
{\displaystyle {\frac {\partial f}{\partial x}}(0,\omega )\in L^{1}(\mu )}
μ
(
Ω
)
<
∞
{\displaystyle \mu (\Omega )<\infty }
|
∂
f
∂
x
(
0
,
ω
)
|
≤
M
{\displaystyle \left|{\frac {\partial f}{\partial x}}(0,\omega )\right|\leq M}
A smooth transition from 0 to nonzero
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The function φ
φ
(
x
)
=
{
e
−
1
x
if
x
>
0
,
0
if
x
≤
0
{\displaystyle \varphi (x)={\begin{cases}e^{-{\frac {1}{x}}}&{\mbox{if }}x>0,\\0&{\mbox{if }}x\leq 0\end{cases}}}
A bump function - a smooth function with compact support
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The function ψ
ψ
(
x
)
=
φ
(
2
(
1
+
x
)
)
φ
(
2
(
1
−
x
)
)
=
{
e
−
1
1
−
x
2
if
|
x
|
<
1
,
0
if
|
x
|
≥
1
{\displaystyle \psi (x)=\varphi \left(2\left(1+x\right)\right)\varphi \left(2\left(1-x\right)\right)={\begin{cases}e^{-{\frac {1}{1-x^{2}}}}&{\mbox{if }}|x|<1,\\0&{\mbox{if }}|x|\geq 1\end{cases}}}
A smooth transition from 0 to 1
edit
This is designed as a partition of unity .
The function η
η
(
x
)
=
φ
(
x
)
φ
(
x
)
+
φ
(
1
−
x
)
=
{
0
if
x
≤
0
,
(
1
+
exp
(
1
−
2
x
x
(
1
−
x
)
)
)
−
1
if
0
<
x
<
1
,
1
if
x
>
1
{\displaystyle \eta (x)={\frac {\varphi (x)}{\varphi (x)+\varphi (1-x)}}={\begin{cases}0&{\mbox{if }}x\leq 0,\\\left(1+\exp \left({\frac {1-2x}{x(1-x)}}\right)\right)^{-1}&{\mbox{if }}0<x<1,\\1&{\mbox{if }}x>1\end{cases}}}
Good-to-know changes of variables
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List of canonical coordinate transformations
Let σd-1 be the uniform probability measure on the d-1 -dimensional unit sphere and let κd be the volume of the d -dimensional unit ball (so that dκd is the surface area of the sphere). Then:
1
d
κ
d
∫
x
∈
R
d
,
R
1
<
|
x
|
<
R
2
f
(
x
)
d
x
=
∫
R
1
R
2
r
d
−
1
(
∫
ζ
∈
R
d
,
|
ζ
|
=
1
f
(
r
ζ
)
d
σ
d
−
1
(
ζ
)
)
d
r
{\displaystyle {\frac {1}{d\kappa _{d}}}\int _{x\in \mathbb {R} ^{d},R_{1}<|x|<R_{2}}f(x)\,dx=\int _{R_{1}}^{R_{2}}r^{d-1}\left(\int _{\zeta \in \mathbb {R} ^{d},\left|\zeta \right|=1}f(r\zeta )\,d\sigma _{d-1}\left(\zeta \right)\right)\,dr}
Corollary: If f is radial , that is: f (x ) = f (|x |), then:
1
d
κ
d
∫
x
∈
R
d
,
R
1
<
|
x
|
<
R
2
f
(
x
)
d
x
=
∫
R
1
R
2
r
d
−
1
f
(
r
)
d
r
{\displaystyle {\frac {1}{d\kappa _{d}}}\int _{x\in \mathbb {R} ^{d},R_{1}<|x|<R_{2}}f(x)\,dx=\int _{R_{1}}^{R_{2}}r^{d-1}f(r)\,dr}
Integral convergence
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This may be proven using the previously-mentioned change of variables.
Supposing ε > 0, we have
∫
x
∈
R
d
,
|
x
|
<
1
1
|
x
|
d
−
ε
d
x
=
d
κ
d
ε
{\displaystyle \int _{x\in \mathbb {R} ^{d},|x|<1}{\frac {1}{|x|^{d-\varepsilon }}}\,dx={\frac {d\kappa _{d}}{\varepsilon }}}
In particular,
∫
x
∈
R
d
,
|
x
|
<
1
1
|
x
|
t
d
x
<
∞
⇔
t
<
d
{\displaystyle \int _{x\in \mathbb {R} ^{d},|x|<1}{\frac {1}{|x|^{t}}}\,dx<\infty \Leftrightarrow t<d}
Let (Ω , P ) be a probability space.
A real-valued random variable is a Borel-measurable
X
:
Ω
→
R
{\displaystyle X:\Omega \to \mathbb {R} }
The expected value of X is
E
[
X
]
=
∫
Ω
X
(
ω
)
d
P
(
ω
)
{\displaystyle \operatorname {E} [X]=\int _{\Omega }X(\omega )\,\mathrm {d} P(\omega )}
Denote by κd the volume of the d -dimensional unit ball . Then
κ
d
=
π
d
/
2
Γ
(
d
2
+
1
)
=
{
π
k
k
!
d
=
2
k
2
k
+
1
π
k
(
2
k
+
1
)
!
!
d
=
2
k
+
1
=
2
,
π
,
4
3
π
,
1
2
π
2
,
8
15
π
2
,
1
6
π
3
,
16
105
π
3
,
…
{\displaystyle \kappa _{d}={\frac {\pi ^{d/2}}{\Gamma \left({\frac {d}{2}}+1\right)}}={\begin{cases}{\dfrac {\pi ^{k}}{k!}}&d=2k\\\\{\dfrac {2^{k+1}\pi ^{k}}{\left(2k+1\right)!!}}&d=2k+1\end{cases}}=2,\pi ,{\frac {4}{3}}\pi ,{\frac {1}{2}}\pi ^{2},{\frac {8}{15}}\pi ^{2},{\frac {1}{6}}\pi ^{3},{\frac {16}{105}}\pi ^{3},\ldots }
Denote by sd-1 the surface area of the d-1 -dimensional unit sphere (the boundary of the d -dimensional unit ball). Then
s
d
−
1
=
d
κ
d
{\displaystyle s_{d-1}=d\kappa _{d}}
Proof .
Let Bd (r) be the d -dimensional Euclidean ball centered at the origin with radius r . Then the following inclusion is true:
[
−
r
d
,
r
d
]
d
⊂
B
d
(
r
)
⊂
[
−
r
,
r
]
d
{\displaystyle \left[-{\frac {r}{\sqrt {d}}},{\frac {r}{\sqrt {d}}}\right]^{d}\subset B^{d}\left(r\right)\subset \left[-r,r\right]^{d}}
(TODO : The more general result with Hölder's inequality, inclusions of Lp spaces, etc.)
![{\displaystyle \operatorname {E} [X]=\int _{\Omega }X(\omega )\,\mathrm {d} P(\omega )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c245c08b20cc61b02758f8f374c965ec101d15a)
![{\displaystyle \left[-{\frac {r}{\sqrt {d}}},{\frac {r}{\sqrt {d}}}\right]^{d}\subset B^{d}\left(r\right)\subset \left[-r,r\right]^{d}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d42c1d3882612aebd2c0bd1cb43f89bc0880a18)