definition 11.34 Measure; measure space
open in the book ·
parts/02-mathematical-methods/09-probability-statistics.tex:883
· p. 381
- ground object -- no derivation owed
Rests on
-
depends_on
definition 11.1
Discrete probability space
¶
-
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 ¶ ↺
-
depends_on
definition 7.45
Series
¶
-
depends_on
definition 11.33
Sigma-algebra
¶
- depends_on definition 11.1 Discrete probability space ¶ ↺
-
depends_on
definition 6.1
Topological space
¶
- depends_on definition 3.31 Empty set ¶
-
depends_on
definition 3.35
Union, intersection, difference
¶
- depends_on definition 3.8 Conjunction ¶
- depends_on definition 3.9 Disjunction ¶
- depends_on definition 3.7 Negation ¶
- depends_on definition 3.30 Subset ¶
- depends_on equation 3.51 eq:set-indexed ¶
Supports
-
depends_on
definition 11.38
Measure-preserving map; invariant measure;
ergodicity
¶
- depends_on theorem 11.39 Birkhoff's pointwise ergodic theorem ¶
-
depends_on
definition 11.35
Null set; almost everywhere
¶
- depends_on definition 11.38 Measure-preserving map; invariant measure; ergodicity ¶ ↺
-
depends_on
proposition 11.36
Monotonicity, countable subadditivity, and null
sets
¶
- depends_on theorem 11.39 Birkhoff's pointwise ergodic theorem ¶ ↺
- depends_on proposition 11.36 Monotonicity, countable subadditivity, and null sets ¶ ↺
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 |
→ | Discrete probability space | declared | parts/02-mathematical-methods/09-probability-statistics.tex:898 |
depends_on |
→ | Sigma-algebra | declared | parts/02-mathematical-methods/09-probability-statistics.tex:898 |
depends_on |
← | Measure-preserving map; invariant measure; ergodicity | declared | parts/02-mathematical-methods/09-probability-statistics.tex:995 |
depends_on |
← | Null set; almost everywhere | declared | parts/02-mathematical-methods/09-probability-statistics.tex:906 |
depends_on |
← | Monotonicity, countable subadditivity, and null sets | declared | parts/02-mathematical-methods/09-probability-statistics.tex:920 |