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I'm on the process of learning measure theory and I have a question

the question is given a finite set and a finite collection of subsets of this set, how we can construct the Smallest σ-algebra containing this collection of subsets.

In other words, is there some kind of an algorithm with which we can construct this σ-algebra based on this collection of subsets.

Abdessamad Jawad
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  • There is no good algorithm. You need to take all complements, then all of their possible unions (here all the unions will be finite), then complements of what you get, then unions again, and so on until you get a sigma algebra. – Mark Sep 24 '21 at 20:58
  • The process is described here – Arturo Magidin Sep 24 '21 at 21:02
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    Note that for finite sets, $\sigma$-algebra is the same thing as algebra, since there are only finitely many possible subsets, and thus there will never be a need to take an infinite union – Arturo Magidin Sep 24 '21 at 21:03
  • @Mark: what is not "good" about your algorithm? I think it is straightforward to implement using the approach of the Warshall closure algorithm. – Rob Arthan Sep 24 '21 at 21:10
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    Imagine you have the throw of a die so ${1,2,3,4,5,6}$ and you are interested in primes ${2,3,5}$ and evens ${2,4,6}$. Then you want to partition the set into those elements which never appear separately $1\mid 2 \mid 3,5 \mid 4,6$ and your sigma algebra will have the $2^4$ elements which combine these four parts – Henry Sep 24 '21 at 21:19

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