Generating the Borel-algebra from half-open intervals

[dane502]

Hi everybody!

I have been asked to show that the Borel-algebra can be generated from the set of half-open intervals of the form [a , b) where a<b.

I know that the set of open intervals of the form (a,b) where a<b generates the Borel-algebra and thought I would go about showing that the to sets generates the same δ-algebra. But that has proven more difficult than I thought.

Can anybody give me an direction?


Any help is greatly appreciated!
dane502

 

[micromass]

Can you write the open intervals as a union of halfopen intervals?

Hi everybody!

I have been asked to show that the Borel-algebra can be generated from the set of half-open intervals of the form [a , b) where a<b.

I know that the set of open intervals of the form (a,b) where a<b generates the Borel-algebra and thought I would go about showing that the to sets generates the same δ-algebra. But that has proven more difficult than I thought.

Can anybody give me an direction?


Any help is greatly appreciated!
dane502

 

[dane502]

Can you write the open intervals as a union of halfopen intervals?

Thank you for your answer.

No, the half-open intervals has to be on the form of [a,b), so any union of those half-open intervals (that is not disjoint) will also have form [a,b).

 

[micromass]

Thank you for your answer.

No, the half-open intervals has to be on the form of [a,b), so any union of those half-open intervals (that is not disjoint) will also have form [a,b).

Sure of that?

 

[micromass]

Think of

$$\bigcup [a-1/n,b)$$

 

[dane502]

Think of

$$\bigcup [a-1/n,b)$$

Thank you. Somehow I missed that.