Infinitely often and almost always: Difference between revisions

Revision as of 21:14, 31 July 2019

Let $A_{1}, A_{2}, A_{3}, \dots$ be a sequence of events in some sample space $Ω$ .

perspective	infinitely often	almost always
unions and intersections	$ω \in ⋂_{N = 1}^{\infty} ⋃_{n = N}^{\infty} A_{n}$	$ω \in ⋃_{N = 1}^{\infty} ⋂_{n = N}^{\infty} A_{n}$
first-order quantifiers	$\forall N \geq 1 \exists n \geq N : ω \in A_{n}$	$\exists N \leq 1 \forall n \geq N : ω \in A_{n}$
verbal expression
lim sup/lim inf	$ω \in {lim sup}_{n \to \infty} A_{n}$	$ω \in {lim inf}_{n \to \infty} A_{n}$
limit of sup/inf	$ω \in lim_{N \to \infty} {sup}_{n = N} A_{n}$	$ω \in lim_{N \to \infty} {inf}_{n = N} A_{n}$

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 | unions and intersections || <math>\omega \in \bigcap_{N=1}^\infty \bigcup_{n=N}^\infty A_n</math> || <math>\omega \in \bigcup_{N=1}^\infty \bigcap_{n=N}^\infty A_n</math>
 |-
-| first-order quantifiers || <math>\forall N\geq 1\ \exists n \geq N:\ \omega \in A_n</math> || <math>\exists N \leq 1\ \forall n \geq N:\ \omega \in A_n</math>
+| first-order quantifiers || <math>\forall N\geq 1\ \exists n \geq N\colon \omega \in A_n</math> || <math>\exists N \leq 1\ \forall n \geq N\colon \omega \in A_n</math>
 |-
 | verbal expression