proposition 7.141 The critical exponent, and the Hausdorff dimension

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:4778 · p. 261

Rests on

depends_on definition 7.140 Hausdorff measure ¶
- 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 3.28 Set ¶
  - depends_on definition 3.38 Ordered pair and Cartesian product ¶
depends_on definition 7.138 Set of measure zero ¶ ↺
depends_on lemma 7.139 A countable union of null sets is null ¶
- depends_on definition 7.138 Set of measure zero ¶ ↺
- depends_on proposition 7.46 Geometric series ¶
  - depends_on corollary A.46 Monotone convergence ¶
    - depends_on theorem A.40 Least-upper-bound property ¶
      - depends_on definition A.36 Cut ¶
      - depends_on definition A.38 Order on $\R$ ¶
      - proves proof app:A-long-proofs@proof-29 ¶
    - proves proof app:A-long-proofs@proof-33 ¶
  - depends_on definition 7.45 Series ¶
    - depends_on definition 7.2 Absolute value ¶
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
  - proves proof ch:05-real-analysis@proof-27 ¶
- depends_on proposition 3.67 Countability of the plane of naturals ¶
  - depends_on definition 3.65 Countable set ¶ ↺
  - depends_on definition 3.47 Bijective map ¶ ↺
  - depends_on definition 3.38 Ordered pair and Cartesian product ¶ ↺
  - proves proof ch:01-logic-sets@proof-13 ¶
- proves proof ch:05-real-analysis@proof-81 ¶
proves proof ch:05-real-analysis@proof-82 ¶

Supports

depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶

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`	→	Hausdorff measure	declared	parts/02-mathematical-methods/05-real-analysis.tex:4792
`depends_on`	→	Set of measure zero	declared	parts/02-mathematical-methods/05-real-analysis.tex:4792
`depends_on`	→	A countable union of null sets is null	declared	parts/02-mathematical-methods/05-real-analysis.tex:4792
`depends_on`	←	The Hausdorff dimension never exceeds the box dimension	declared	parts/02-mathematical-methods/05-real-analysis.tex:4883
`proves`	←	ch:05-real-analysis@proof-82	declared	parts/02-mathematical-methods/05-real-analysis.tex:4795