proposition 11.36 Monotonicity, countable subadditivity, and null sets

open in the book · parts/02-mathematical-methods/09-probability-statistics.tex:909 · p. 381

Rests on

depends_on definition 11.34 Measure; measure space ¶
- 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 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 ¶
depends_on definition 11.35 Null set; almost everywhere ¶
- depends_on definition 11.34 Measure; measure space ¶ ↺
proves proof ch:09-probability-statistics@proof-14 ¶

Every logical edge within two steps of this node.

type	direction	node	provenance	where
`depends_on`	→	Measure; measure space	declared	parts/02-mathematical-methods/09-probability-statistics.tex:920
`depends_on`	→	Null set; almost everywhere	declared	parts/02-mathematical-methods/09-probability-statistics.tex:920
`depends_on`	←	Birkhoff's pointwise ergodic theorem	declared	parts/02-mathematical-methods/09-probability-statistics.tex:1013
`proves`	←	ch:09-probability-statistics@proof-14	declared	parts/02-mathematical-methods/09-probability-statistics.tex:923