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Find smallest $sigma$-algebra with intersections

Mathematics Asked by Puigi on December 17, 2020

I was asked to find the smallest $sigma$-algebra ($sigma(Z)$) generated by the following set $Z$={{1,2},{2,3},{3,4}} where the sample space is $Omega$={1,2,3,4}. The thing is that the middle element of $Z$ is causing me a lot of trouble. The smallest $sigma(Z)$ I could find was the following:

$sigma(Z)$={$emptyset$,$Omega$,{1},{2},{3},{4},{1,2},{2,3},{3,4},{1,3},{1,4},{2,4},{1,2,3},{2,3,4},{1,3,4},{1,2,4}}

I know that the $sigma$-algebra has to be closed under complements and unions, and that is how I got to that set. Is this right? Is there a faster way to solve this kind of problems?

2 Answers

The moment you get ${1},{2},{3},{4}$ in the sigma algebra you can conclude that all subsets of ${1,2,3,4}$ are also in it. (Because any subset of ${1,2,3,4}$ is a finite union of singletons). In this case ${1}={1,2,}setminus {2,3}$, ${2}={1,2}cap {2,3}$, ${3}={2,3,}setminus ({1,2}$ and ${4}={3,4}setminus ({2,3}$.

Correct answer by Kavi Rama Murthy on December 17, 2020

There is a faster way : Since ${1},{2},{3},{4}in sigma (Z)$, then $sigma (Z)$ is the power set of ${1,2,3,4},$ and what you found is correct.

Answered by Surb on December 17, 2020

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