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Math Help - Borel sets

  1. #1
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    Borel sets

    How do I prove the following?

    If X is a Borel set, and a \in \mathbb{R}, prove that X+a=\{x+a, x \in X\} is also a Borel set.

    Thank you!
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  2. #2
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    Quote Originally Posted by harriette View Post
    How do I prove the following?

    If X is a Borel set, and a \in \mathbb{R}, prove that X+a=\{x+a, x \in X\} is also a Borel set.

    Thank you!
    Fix a, and use a monotone class argument. So for any open set B, consider B+a, which generate X+a. It is rather obvious that all the open sets are of the form B+a; If you give me a set N open, N-a is open, and so N=(N-a)+a. Hence the two sigma algebras coincide.
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  3. #3
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    Quote Originally Posted by Focus View Post
    Fix a, and use a monotone class argument. So for any open set B, consider B+a, which generate X+a. It is rather obvious that all the open sets are of the form B+a; If you give me a set N open, N-a is open, and so N=(N-a)+a. Hence the two sigma algebras coincide.
    Thank you for your time!
    Could you please tell me what a monotone class argument is?
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  4. #4
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    Quote Originally Posted by harriette View Post
    Thank you for your time!
    Could you please tell me what a monotone class argument is?
    When you show that the generators (pi systems) are the same. So in this case the Borel sets are generated by the open sets. As \cup(A_i+a)=\cup(A_i)+a, the sets of the form X+a are generated by U+a where U is open. If you know that the generators are the same, then you know that the sigma algebra they generate are the same as well.
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  5. #5
    Member mabruka's Avatar
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    In other words use the Monotone Class theorem, also known as Dynkin's lemma or the \pi-\lambda theorem.
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