Infinitely often and almost always: Difference between revisions

Revision as of 21:16, 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	$ω \in A_{n}$ for infinitely many $n \geq 1$	$ω \in A_{n}$ for almost all $n \geq 1$ , i.e. $ω \in A_{n}$ for all but finitely many $n \geq 1$ , i.e. $ω \notin A_{n}$ for finitely many $n \geq 1$
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	$ω \in lim_{N \to \infty} {sup}_{n \geq N} A_{n}$	$ω \in lim_{N \to \infty} {inf}_{n \geq N} A_{n}$

@@ Line 9: / Line 9: @@
 | 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
+| verbal expression || <math>\omega \in A_n</math> for infinitely many <math>n\geq 1</math> || <math>\omega \in A_n</math> for almost all <math>n\geq 1</math>, i.e. <math>\omega \in A_n</math> for all but finitely many <math>n \geq 1</math>, i.e. <math>\omega \notin A_n</math> for finitely many <math>n \geq 1</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>\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>
+| limit of sup/inf || <math>\omega \in \lim_{N\to\infty} \sup_{n\geq N} A_n</math> || <math>\omega \in \lim_{N\to\infty} \inf_{n\geq N} A_n</math>
 |}