The chapter on measurable spaces introduces a -algebra over the real numbers . The Borel algebra, , is the smallest -algebra where is the algebra generated by finite disjoint sums of intervals of the form . By the direct product of algebras we also get algebras over higher dimensions . We also get a legal -algebra for the infinite direct product .
The book asks to show that certain sets are members of . Show that the following are Borel sets.
Take the first case. Note that is not satisfied if for every , . This can only happen if there are an infinite number of coordinates whose value is . Let
The set is a Borel set since we have constructed it as a countable union of Borel sets. Therefore, (this is also a countable union) is a Borel set. A similar argument is made for the other.