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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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.

EreneDreaceaw · Accepted Answer

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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