the $\sigma$ algeba generated by the class of open intervals with rational end points coincide with the borel $\sigma$ algebra on the real line.

Hint: Clearly the $\sigma$ algebra generated by open intervals with rational end points is contained in the Borel $\sigma$ algebra. So now you only need to show one thing:

• Any open interval can be written as a countable union of rational end point open intervals

Then you can show that any Borel $\sigma$ algebra set can be obtained through $\sigma$ algebra operations on the collection of rational endpoint open sets.

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Updated on September 29, 2022