lemma 7.139 A countable union of null sets is null
open in the book ·
parts/02-mathematical-methods/05-real-analysis.tex:4728
· p. 261
Rests on
-
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 7.125
Multiple integral
¶
-
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
theorem A.40
Least-upper-bound property
¶
-
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
corollary A.46
Monotone convergence
¶
-
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 ¶
Supports
- depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶
-
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 7.94 Three numbers, two kinds ¶
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 |
→ | Set of measure zero | declared | parts/02-mathematical-methods/05-real-analysis.tex:4733 |
depends_on |
→ | Geometric series | declared | parts/02-mathematical-methods/05-real-analysis.tex:4733 |
depends_on |
→ | Countability of the plane of naturals | declared | parts/02-mathematical-methods/05-real-analysis.tex:4733 |
depends_on |
← | The Hausdorff dimension never exceeds the box dimension | declared | parts/02-mathematical-methods/05-real-analysis.tex:4883 |
depends_on |
← | The critical exponent, and the Hausdorff dimension | declared | parts/02-mathematical-methods/05-real-analysis.tex:4792 |
depends_on |
← | Three numbers, two kinds | declared | parts/02-mathematical-methods/05-real-analysis.tex:2849 |
proves |
← | ch:05-real-analysis@proof-81 | declared | parts/02-mathematical-methods/05-real-analysis.tex:4736 |