Explore BrainMass

Explore BrainMass

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

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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.

    © BrainMass Inc. brainmass.com October 10, 2019, 1:37 am ad1c9bdddf

    Solution Preview

    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 ...

    Solution Summary

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