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Alannah Short 2022-06-26 Answered
Let ( X , A ) y ( Y , B ) measure spaces. Let R : = R ( A , B ) the collection of measurable rectangles R : = { A × B : A A , B B } .. Prove that the algebra of subsets of X × Y generated by R is the collection of finite unions of elements of R. I have a hint and it is in which I call C to be the collection of finite unions of elements of R, prove that C is in the algebra generated by R, then prove that C is an algebra that contains R.
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Answers (1)

EreneDreaceaw
Answered 2022-06-27 Author has 20 answers
You only need to prove that the set A of finite disjoint unions of elements of R is an algebra.
Note that R has the following properties:
R ,
E , F R E F R ,
E R E c  is a finite disjoint union of members of  R .
Any set R that satisfies the above three hypotheses has the property that the algebra generated by R is the collection of finite disjoint unions of elements of R. This is not difficult to verify.

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