Analysis
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Fubini's theorem and Tonelli's theorem
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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\}} be a measurable function.
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 } provided one of the following criteria:
(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 )} then f n → f {\displaystyle f_{n}\to f} in L 1 (μ ) by Scheffé’s lemma .
(Dominated convergence theorem ) | f n | ≤ g {\displaystyle \left|f_{n}\right|\leq g} for some g ∈ L 1 ( μ ) {\displaystyle g\in L^{1}\left(\mu \right)} (everywhere, or μ -almost everywhere if μ is a complete measure ).
Note: This also gives us f n → f {\displaystyle f_{n}\to f} in 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 } and | f n | ≤ M {\displaystyle \left|f_{n}\right|\leq M} .
Note: This also gives us f n → f {\displaystyle f_{n}\to f} in 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 )} , wherein x ∈ R {\displaystyle x\in \mathbb {R} } , and if 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 )} provided one of the following criteria:
∂ 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 } and | ∂ f ∂ x ( 0 , ω ) | ≤ M {\displaystyle \left|{\frac {\partial f}{\partial x}}(0,\omega )\right|\leq M} .Smooth functions
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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
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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}}}
Calculus
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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} .
Probability
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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 )} . Geometry
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Euclidean balls
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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.)