Title: Theory / Unimportant / Math / Measure theory - Fremlin{ / 22 Real Analysis
Tags: hasSeenConjDef fromImplication EquivalentButNoIntermediate addIntermediateCard fromMath integralprimitive oldTypToDelete riemanIntegralOfF récupéré

Smaller0: for each \(x\in A\) there exists \(x\in I\in\mathcal I\) with \(I\) arbitrarily small
Greater0: there exists a countable disjoint set \(\mathcal I_0\subseteq\mathcal I\) such that \(\mu(A\setminus\bigcup\mathcal I_0)=0\), with \(\mu\) Lebesgue's measure
Page: 55
Chapter: 22
Section: 1A
Variables: \(A\subseteq\mathbb R\) bounded,
\(\mathcal I\) a family of non-singleton closed intervals
Name: Vitali's theorem in \(\mathbb R\)

Name: A disjoint set
Definition: A set of pair-wise disjoint sets
Page: 55
Chapter: 22
Section: 1A

Question: How to choose a family \(\mathcal I\) as in Vitali's theorem, such that \(\mu(A\setminus \bigcup_{i\in\mathcal I}I)=0\) ?
Answer: -Either \(A=\emptyset\), in this case \(\mathcal I=\emptyset\)
-Either \(A=\bigcup I_i\), in this case take \(\mathcal I=\{I_i\}\)
-Or construct recursively an increasing sequence \(\mathcal I_i\), starting empty. Let \(\mathcal J_i\) the set of intervals, intersecting \(A\), disjoint from \(\bigcup_{I\in\mathcal I_{i-1}}I\). Let \(\gamma_n\) the supremum of the lengths of the elements of \(\mathcal J_i\).
Adding to \(\mathcal I_i\) an element \(I\), disjoint from \(\bigcup_{I\in\mathcal I_{i-1}}I\), whose length is at least \(f(n)\gamma_n\), with \(f(n)\in[\frac12,1)\)
Page: 55, 56
Chapter: 22
Section: 1A, (b)
Variables: \(A\subseteq\mathbb R\) bounded,
\(\mathcal I\) a family of non-singleton closed intervals
Assuming: \(x\in A\) implies \(x\in I\in\mathcal I\) with \(I\) arbitrarily small

Smaller0: \(f\) monotonic
Greater0: \(f\) differentiable a.e.
Page: 58
Chapter: 22
Section: 2A
Variables: \(I\subseteq\mathbb R\) an interval,
\(f:I\to\mathbb R\)

Name: The indefinite integral of \(f\)
Page: 183
55
Chapter: 3
22 (Fremlin's measure theory)
Section: 2
2C
Variables: \(f:[a,b]\to \mathbb R\) Riemann integrable
Notation: [$]\int_a^xf(t)\dd t[/$]
Index: 34
Kind: Cor
Lem
Typ: Continuous
Denoted by: \(F\)

Definition: \(\text{sup}_{L\subseteq K\text{ finite}}\sum_{k\in L}a_k\)
Page: 59
Chapter: 22
Section: B
Variables: \(K\) a set,
\((a_k)_{k\in K}\in\mathbb R^{\ge0}\)
Context: K potentially infinite
Notation: \(\sum_{k\in K}a_k\)
Kind: Rem

Smaller0: \(F\) non-decreasing
Greater0: exists and is at most \(F(b)-F(a)\)
Page: 59
Chapter: 22
Section: 2C
Variables: \(a\le b\in\mathbb R\),
\(F:[a,b]\to\mathbb R\)
Kind: Lem
Prefix Greater: \(\int_a^bF'\)

Question: [$]\int_x^y F\dd\mu+\int_y^zF\dd\mu[/$]
Answer: equals [$]\int_x^z F\dd\mu[/$]
Page: 60
Chapter: 22
Section: 2C
Variables: \(a\le x\le y\le b\in\mathbb R\),
\(F:[a,b]\to\mathbb R\),
\(\mu\) a measure on \([a,b]\)
Kind: Lem
Assuming: \(\mu\) is 0 on singleton

Smaller0: \(\int_a^x f=\int_a^xg\) for every \(x\in[a,b]\)
Greater0: \(\int_E f=\int_Eg\) for every \(E\subseteq[a,b]\) measurable
Page: 60
Chapter: 22
Section: 2D
Variables: \(a\le b\in\mathbb R\),
\(f,g:[a,b]\to\mathbb R\),
Kind: Lem
Smaller to greater: iff
Greatest: \(f=g\) a.e.

Smaller0: \(f\) is integrable over \([a,b]\)
Greater0: exists and equal to \(f(x)\) a.e.
Page: 62, 63
Chapter: 22
Section: 2H
Variables: \(-\infty\le a<x<\le b\le\infty\),
\(f:[a,b]\to\mathbb C\),
Kind: Lem, Cor
Prefix Greater: [$]\derivative{\int_{E\cap]-\infty,x[}f(t)\dd t}{x}[/$]
Smaller1: \(f\) is continuous at \(x\)

Smaller0: \(f\) is integrable over \(E\)
Greater0: exists and equal to \(f(x)\) a.e.
Page: 61-63
Chapter: 22
Section: 2E, F, G
Variables: \(E\subseteq \mathbb R\),
\(f:E\to\mathbb C\)
Kind: Lem, Cor
Prefix Greater: [$]\derivative{\int_{E\cap]-\infty,x[}f}{x}[/$]
Smaller1: \(E\) Lebesgue measurable

Definition: [$]\int_{[x,y)}F\dd\mu[/$]
Page: 60
Chapter: 22
Section: 2C
Variables: \(a\le x\le y\le b\in\mathbb R\),
\(F:[a,b]\to\mathbb R\),
\(\mu\) a measure on \([a,b]\)
Notation: [$]\int_x^y F\dd\mu[/$]
Kind: Lem
Assuming: \(\mu\) is 0 on singleton

Definition: arbitrarily close to \(x\)
Page: 63
Chapter: 22
Section: 2J,K
Variables: \(A\subseteq\mathbb R\)
Notation: \(\tilde A^-\)
Definition2: strictly smaller than \(x\)
Conjdef: the set of \(x\in A\) such that \(A\) contains number
Cardinal: a cocountable subset of \(A\)

Definition: arbitrarily close to \(x\)
Page: 63
Chapter: 22
Section: 2J,K
Variables: \(A\subseteq\mathbb R\)
Notation: \(\tilde A^+\)
Definition2: strictly greater than \(x\)
Conjdef: the set of \(x\in A\) such that \(A\) contains number
Cardinal: a cocountable subset of \(A\)

Name: The Dini derivatives of \(f\)
Definition: \(x\) belonging to \(\tilde A^+\) or \(\tilde A^-\) (according to the symbol in the notation)
Page: 63
Chapter: 22
Section: 2J
Variables: \(A\subseteq\mathbb R\)
\(f:A\to\mathbb R\)
Notation: \(\overline D^+\), \(\underline D^+\), \(\overline D^-\), \(\underline D^-\),
Definition2: to the extended real which is the superior limit (respectively inferior limit) for \(D\) or \(\overline D\)(respectively for \(d\), \(\underline D\))
Conjdef: The function sending
Definition3: when \(y\) tends toward \(x\) by above (respectively below) when the symbol is \(+\), (respectively \(-\))
Definition4: of \(\frac{f(y)-f(x)}{y-x}\)
Notation2: \(D^+\), \(d^+\), \(D^-\), \(d^-\)

Definition: [$]\defFun{\tilde A^-}{[-\infty,\infty]}{x}{\text{lim sup}_{y\in A,y\uparrow x}\frac{f(y)-f(x)}{y-x}}[/$]
Page: 63
Chapter: 22
Section: 2J
Variables: \(A\subseteq\mathbb R\)
\(f:A\to\mathbb R\),
Notation: \(\overline D^-\)
Notation2: \(d^-\)

Definition: [$]\defFun{\tilde A^+}{[-\infty,\infty]}{x}{\text{lim inf}_{y\in A,y\downarrow x}\frac{f(y)-f(x)}{y-x}}[/$]
Page: 63
Chapter: 22
Section: 2J
Variables: \(A\subseteq\mathbb R\)
\(f:A\to\mathbb R\),
Notation: \(\underline D^+\)
Notation2: \(d^+\)

Definition: [$]\defFun{\tilde A^+}{[-\infty,\infty]}{x}{\text{lim sup}_{y\in A,y\downarrow x}\frac{f(y)-f(x)}{y-x}}[/$]
Page: 63
Chapter: 22
Section: 2J
Variables: \(A\subseteq\mathbb R\)
\(f:A\to\mathbb R\)
Notation: [$]\overline D^+[/$]
Notation2: \(D^+\)

Definition: [$]\defFun{\tilde A^-}{[-\infty,\infty]}{x}{\text{lim inf}_{y\in A,y\uparrow x}\frac{f(y)-f(x)}{y-x}}[/$]
Page: 63
Chapter: 22
Section: 2J
Variables: \(A\subseteq\mathbb R\)
\(f:A\to\mathbb R\),
Notation: \(\underline D^-\)
Denoted by: \(d^-\)

Smaller0: \(\underline D^- f\)
Greater0: \(\overline D^-f\)
Page: 63
Chapter: 22
Section: 2J
Variables: \(A\subseteq\mathbb R\)
\(f:A\to\mathbb R\),
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Definition: \(\tilde A^+\cap\tilde A^-\)
Page: 63
Chapter: 22
Section: 2L
Variables: \(A\subseteq\mathbb R\)
Notation: \(\tilde A\)
Cardinal: a cocountable subset of \(A\)

Smaller0: equal and finite
Greater0: (\((\overline D^\pm f)(x)=+\infty\) and \((\underline D^\pm f)(x)=-\infty\))
Page: 63
Chapter: 22
Section: 2L
Variables: \(A\subseteq\mathbb R\)
\(f:A\to\mathbb R\),
\(x\in A\)
Smaller to greater: or
Smaller1: \((\underline D^+f)(x)=-\infty\) and \((\overline D^-f)(x)=+\infty\)
Name: The Denjoy-Young-Saks theorem
Prefix Smaller: (\((\overline D^+f)(x)\) and \((\underline D^-f)(x)\) are
Smallest: (each Dini derivates exists, are finite and equal)
Greatest: \(x\) belongs to some fixed negligeable set
Intermediate: the same by symmetry

Question: the set of \(x\) such that \((\underline D^\pm)=+\infty\) is
Answer: negligible
Page: 66
Chapter: 22
Section: 2 notes
Variables: \(A\subseteq\mathbb R\)
\(f:A\to\mathbb R\),

Question: \[\lim_{h\downarrow 0}\frac{1}{h}\int_x^{x+h}f\]
Answer: equals \(f(x)\) for almost every \(x\in I\)
Page: 66
Chapter: 22
Section: 3A
Variables: \(I\) an interval in \(\mathbb R\)
\(f:I\to\mathbb C\)
Assuming: \(f\) integrable over \(I\)
Name: Lebesgue’s Density Theorem: integral form
Construction: Let \(F(x)=\int_{I\cap]-\infty,x[}f\)
\(f(x)=F'(x)=\lim_{h\downarrow0}\frac1h(F(x+h)-F(x))=\lim_{h\downarrow0}\frac1h\int_{x}^{x+h}f\)

Question: \[\lim_{h\downarrow 0}\frac{1}{2h}\mu(E\cap [x-h,x+h])\]
Answer: equals \(1\) (resp. 0) for almost every \(x\) belonging (resp. not belonging) to \(E\),
Page: 66
Chapter: 22
Section: 3
Variables: \(E\subseteq \mathbb R\)
Index: B
Kind: Cor
Assuming: \(E\) measurable

Name: A density point of \(E\)
Definition: A point \(x\in \mathbb R\) such that \[\lim_{h\downarrow 0}\frac{1}{2h}\mu(E\cap [x-h,x+h])=1\]
Page: 67
Chapter: 22
Section: 3
Variables: \(E\subseteq\mathbb R\)
Index: B
Kind: Cor

Question: [$$]\lim_{h\downarrow 0}\frac{1}{2h}\mu\{y\in\mathrm{dom}f\mid \abs{y-x}\le h,\abs{f(y)-f(x)}\le\epsilon\}[/$$]
Answer: equals 1 for almost every \(x\)
Page: 67
Chapter: 22
Section: 3
Variables: \(E\subseteq \mathbb R\)
\(\epsilon>0\)
\(f:\mathbb R\to\mathbb C\)
Index: C
Kind: Cor
Assuming: \(f\) integrable

Question: [$$]\lim_{h\downarrow 0}\frac{1}{2h}\mu\{y\in\mathrm{dom}f\mid \abs{y-x}\le h,\abs{f(y)-f(x)}\ge\epsilon\}[/$$]
Answer: equals 0 for almost every \(x\)
Page: 67
Chapter: 22
Section: 3
Variables: \(E\subseteq \mathbb R\)
\(\epsilon>0\)
\(f:\mathbb R\to\mathbb C\)
Index: C
Kind: Cor
Assuming: \(f\) integrable

Question: [$$]\lim_{h\downarrow 0}\frac{1}{2h}\int_{x-h}^{x+h}\abs{f(y)-f(x)}\dd y[/$$]
Answer: equals 0 for almost every \(x\in I\)
Page: 67
Chapter: 22
Section: 3
Variables: \(I\) an interval of reals
\(f:\mathbb R\to\mathbb C\)
Index: D
Kind: Cor

Name: The Lebesgue set of \(f\)
Definition: the set of reals \(x\) such that [$$]\lim_{h\downarrow 0}\frac{1}{2h}\int_{x-h}^{x+h}\abs{f(y)-f(x)}\dd y=0[/$$]
Page: 68
Chapter: 22
Section: 3
Variables: \(f:\mathbb R\to\mathbb C\)
Index: D
Kind: Cor
Intuition: The set of point where \(f(x)\) is well approximated by \(f(y)\) for \(y\) in the neighborhood of \(x\).