How do I prove the following?

If X is a Borel set, and , prove that is also a Borel set.

Thank you!

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- March 8th 2010, 05:02 PMharrietteBorel sets
How do I prove the following?

If X is a Borel set, and , prove that is also a Borel set.

Thank you! - March 8th 2010, 05:53 PMFocus
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.

- March 11th 2010, 05:02 PMharriette
- March 11th 2010, 05:33 PMFocus
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 , 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.

- March 11th 2010, 05:38 PMmabruka
In other words use the Monotone Class theorem, also known as Dynkin's lemma or the theorem.