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	$ω \in lim_{N \to \infty} {sup}_{n = N} A_{n}$	$ω \in lim_{N \to \infty} {inf}_{n = N} A_{n}$

@@ Line 5: / Line 5: @@
 ! perspective !! infinitely often !! almost always
 |-
-| unions and intersections || <math>x \in \bigcap_{N=1}^\infty \bigcup_{n=N}^\infty A_n</math> || <math>x \in \bigcup_{N=1}^\infty \bigcap_{n=N}^\infty A_n</math>
+| 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 \ x \in A_n</math> || <math>\exists N \leq 1\ \forall n \geq N\ x \in 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>
 |-
 | verbal expression
 |-
-| lim sup/lim inf || <math>x \in \limsup_{n\to\infty} A_n</math> || <math>x \in \liminf_{n\to\infty} A_n</math>
+| lim sup/lim inf || <math>\omega \in \limsup_{n\to\infty} A_n</math> || <math>\omega \in \liminf_{n\to\infty} A_n</math>
 |-
-| limit of sup/inf || <math>x \in \lim_{N\to\infty} \sup_{n=N} A_n</math> || <math>x \in \lim_{N\to\infty} \inf_{n=N} A_n</math>
+| limit of sup/inf || <math>\omega \in \lim_{N\to\infty} \sup_{n=N} A_n</math> || <math>\omega \in \lim_{N\to\infty} \inf_{n=N} A_n</math>
 |}