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

Question

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

I tried to solve the question but I cannot do proper solution, thus i cannot write here. please show me this question. thank you.

Answer 1

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.