definition 7.140 Hausdorff measure

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:4752 · p. 261

ground object -- no derivation owed

Rests on

depends_on axiom 7.1 Completeness of $\R$ ¶
depends_on definition 7.138 Set of measure zero ¶
- depends_on definition 7.125 Multiple integral ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
  - depends_on definition 7.39 Darboux sums and the definite integral ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
- depends_on definition 3.65 Countable set ¶
  - depends_on axiom 3.33 Peano axioms ¶
  - depends_on definition 3.62 Equipotence ¶
    - depends_on definition 3.47 Bijective map ¶
    - depends_on definition 3.40 Cardinal ¶
depends_on definition 6.24 Metric ¶
- depends_on definition 3.43 Map ¶
  - depends_on definition 3.24 Quantifiers ¶
    - depends_on definition 3.23 Predicate ¶
      - depends_on definition 3.1 Logical proposition ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - depends_on definition 3.28 Set ¶
- depends_on definition 3.38 Ordered pair and Cartesian product ¶ ↺

Supports

depends_on definition 7.142 Box-counting dimension ¶
- depends_on definition 32.79 Box-counting dimension, restated from Part II ¶
  - depends_on definition 32.81 Correlation dimension ¶
  - depends_on example 32.80 The middle-thirds Cantor set ¶
- depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶
- depends_on remark 7.144 What is developed here, and what is not ¶
depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶
- depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶ ↺
depends_on remark 7.144 What is developed here, and what is not ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Completeness of $\R$	declared	parts/02-mathematical-methods/05-real-analysis.tex:4775
`depends_on`	→	Set of measure zero	declared	parts/02-mathematical-methods/05-real-analysis.tex:4775
`depends_on`	→	Metric	declared	parts/02-mathematical-methods/05-real-analysis.tex:4775
`depends_on`	←	Box-counting dimension	declared	parts/02-mathematical-methods/05-real-analysis.tex:4871
`depends_on`	←	The critical exponent, and the Hausdorff dimension	declared	parts/02-mathematical-methods/05-real-analysis.tex:4792
`depends_on`	←	What is developed here, and what is not	declared	parts/02-mathematical-methods/05-real-analysis.tex:4939