Alphabeta MathSigma Algebras and Borel Sets
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1 · Prerequisites
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- Topology of ℝ
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
2 · Logical flowchart
3 · Items
Definition
Algebras of subsets
Definition
Sigma-algebras
Definition
Measurable spaces and measurable sets
Definition
The sigma-algebra generated by a family of sets
Theorem
Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal
Definition
Pi-systems
Definition
Lambda-systems, or Dynkin systems
Proposition
A family is a lambda-system exactly when it contains X and is closed under complements and countable disjoint unions
Definition
The lambda-system generated by a family of sets
Lemma
The generated lambda-system exists and is minimal
Definition
Monotone classes of sets
Definition
The monotone class generated by a family of sets
Lemma
The generated monotone class exists and is minimal
Definition
The Borel sigma-algebra of a topological space
Definition
The trace of a sigma-algebra on a subset
Definition
Limit superior and limit inferior of a sequence of sets
Theorem
Sigma-algebras are closed under countable intersections, differences, symmetric differences, and set limits
Proposition
Set liminf means eventual membership, set limsup means repeated membership, and liminf is contained in limsup
Theorem
Generated sigma-algebras are monotone in their generators and idempotent
Theorem
Two families generate the same sigma-algebra when each lies in the sigma-algebra generated by the other
Lemma
An algebra closed under countable disjoint unions is a sigma-algebra
Lemma
An algebra closed under increasing countable unions is a sigma-algebra
Lemma
A lambda-system closed under finite intersections is a sigma-algebra
Lemma
For a member A of a lambda-system D, the sets B with A intersection B in D form a lambda-system
Lemma
The lambda-system generated by a pi-system is closed under finite intersections
Theorem
Dynkin's pi-lambda theorem
Lemma
The monotone class generated by an algebra is closed under complements
Lemma
Every member of the generated monotone class intersects every original algebra member inside the generated class
Lemma
The monotone class generated by an algebra is closed under finite intersections
Theorem
The monotone class generated by an algebra equals the sigma-algebra it generates
Proposition
Every sigma-algebra is a lambda-system and a monotone class
Theorem
The trace of a sigma-algebra is a sigma-algebra on the traced subset
Theorem
Generating a sigma-algebra commutes with taking traces
Lemma
Every open subset of the real line is a countable union of open intervals with rational endpoints
Theorem
Seven generating families for the Borel sigma-algebra on the real line
Theorem
For n at least one, open sets, closed sets, compact sets, open balls, boxes, rational open boxes, and rational half-open boxes generate the Borel sigma-algebra on ℝⁿ
Theorem
The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra
Theorem
A continuous map has Borel preimages of Borel sets
Theorem
Assuming countable choice, a generated sigma-algebra is obtained in omega-one stages of complements and countable unions
Theorem
Assuming the Axiom of Choice, an infinite family E generates at most |E|ᵃleph-zero sets
Theorem
Assuming the Axiom of Choice, the Borel sigma-algebra on ℝⁿ has cardinality continuum for n at least one
Lemma
A sigma-algebra with a listed infinite subfamily contains a disjoint sequence of nonempty members
Theorem
Assuming countable choice, every infinite sigma-algebra contains a copy of the power set of the natural numbers
Corollary
No sigma-algebra is countably infinite
Theorem
A countable partition generates exactly the unions of its blocks, and the resulting sigma-algebra is countable exactly for a finite partition
Sources
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, Definition 2.1
- T. Tao, An Introduction to Measure Theory, Definition 1.4.12
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, Lemma 2.7
- T. Tao, An Introduction to Measure Theory, Definition 1.4.14
- T. Tao, An Introduction to Measure Theory, Exercises 1.4.13-1.4.14
- A. Dembo, Probability Theory lecture notes, Definition 1.1.36
- A. Dembo, Probability Theory lecture notes, Definition 1.1.36 and the Remark following it
- A. Dembo, Probability Theory lecture notes, proof of Theorem 1.1.38
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, Definition 2.9
- A. Dembo, Probability Theory lecture notes, Definition 1.1.43
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, Definition 2.9 and Theorem 2.10
- T. Tao, An Introduction to Measure Theory, Definition 1.4.16
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, Section 2.1
- T. Tao, An Introduction to Measure Theory, Exercise 1.4.12
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, Exercise 2.9
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, Definition 2.1 and Exercise 2.9
- T. Tao, An Introduction to Measure Theory, Exercise 1.4.14
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, proof route for Theorem 2.10
- A. Dembo, Probability Theory lecture notes, Proposition 1.1.37
- A. Dembo, Probability Theory lecture notes, Theorem 1.1.38
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, proof of Theorem 2.10
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, Theorem 2.10
- A. Dembo, Probability Theory lecture notes, Theorem 1.1.44
- A. Dembo, Probability Theory lecture notes, Definitions 1.1.36 and 1.1.43
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, Proposition 2.8
- T. Tao, An Introduction to Measure Theory, Remark 1.4.15
- T. Tao, An Introduction to Measure Theory, Exercise 1.4.15
- D. H. Fremlin, Measure Theory, Chapter 56, result 567E(b)
- T. Tao, An Introduction to Measure Theory, Exercise 1.4.16
- D. H. Fremlin, Measure Theory, Chapter 56, Section 561A and result 567E(b)
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, Exercises 2.6 and 2.8
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, Exercise 2.6
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, Exercise 2.8
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, Examples 2.4-2.6
- T. Tao, An Introduction to Measure Theory, Exercise 1.4.10