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.

Do you mean rational END points? You originally said in the title and your text "rational and points" — user2566092, Mar 04 '15 at 22:39
The linked duplicate asked a more open ended question, but its answer includes the answer to this. — Jonas Meyer, Mar 06 '15 at 20:37

score 4 · Accepted Answer · answered Mar 04 '15 at 22:43

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.

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

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