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Prove that a given collection of sets is a sigma-algebra

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Let A be a sigma algebra of subsets of R (Real numbers) and suppose I is a closed interval which is in A. Let A(I) denote the collection of all subsets of I which are in A. prove that A(I) is a sigma algebra of subsets of I.

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Solution Summary

A direct proof that the given sollection of intervals is a sigma-algebra is presented.

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To prove that A(I) is a sigma-algebra, we need to show that the following three axioms are satisfied:

1) A(I) is not an empty collection of sets.
2) If X belongs to A(I), then the compliment X' also belongs to A(I). (A(I) is closed under complementation).
3) A(I) is closed under countable unions, that is, if X_1, X_2, X_3,... are in A(I), then the union U of these sets is in ...

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