Alphabeta MathHaar Measure Existence and Uniqueness
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Measure Theory
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- limsup, liminf, and Subsequential Limits
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Left Haar integral and left Haar measure
Lemma
Translations preserve compactly supported continuous functions
Definition
Haar covering ratio of test functions
Lemma
Haar covering ratios are finite and positive
Lemma
Normalized approximate Haar functionals are positive and invariant in the limit
Lemma
Haar covering functionals are asymptotically additive
Lemma
Haar candidate sets have the finite intersection property
Theorem
Existence of a left Haar integral
Corollary
Existence of left and right Haar measures
Lemma
Haar measure is positive on nonempty open sets and finite on compact sets
Lemma
Compactly supported kernels admit commuting radon integrals
Lemma
Haar integral comparison inequality
Theorem
Uniqueness of left Haar measure up to scale
Corollary