Useful pi systems generating the borel sigma algebra

[stukbv]

Homework Statement

For the purpose of my module, we define the lebesgue measure over Int(a,b) where
Int(a,b) = (a,b] where -∞≤a≤b<∞
(a,∞) else
And l((Int(a,b) = b-a a≤b<∞
∞ a≤b =∞

And thus we say the borel sigma algebra is generated by C={Int(a,b) -∞≤a≤b≤∞}
We then get told that a variety of pi systems generate the borel sigma algebra, other than C={Int(a,b) -∞≤a≤b≤∞} ;
1. C' = {(a,b] -∞<a≤b<∞}
2. C_0 = {(a,b) -∞≤a≤b<∞ }
3. C_open = { A [itex]\subseteq[/itex] Reals : A open}
4 C_closed = { B [itex]\subseteq[/itex] Reals: B closed }
5. C_half = {(-∞,b] b [itex]\in[/itex] Reals }

We need to prove that 1-5 generate the borel sigma algebra and that lebesgue is sigma finite on 1-4 but not 5 (to do this I need to find an increasing sequence of events in each C such that the union is the sample space (the reals) and that l(Ai) < ∞ for all i .

The Attempt at a Solution

Basically I have proved 1-4 generate the borel sigma algebra, but 5 is proving difficult, I know that C_half [itex]\subseteq[/itex] C [itex]\subseteq[/itex] σ(C) so this proves that
σ(C_half) [itex]\subseteq[/itex]σ(C), but now I need to show it the other way round, to show that they're equal. I can't seem to figure this out.

Secondly, again I have shown sigma finite for 1 and 2, but i can't do 3 (and therefore 4), with all the others I did how I had been taught, i.e for 2 I said let Ai = (-i,i) then Ai is increasing, the union is the reals and then I wrote (-i,i) = U{n=1..} (-i,i-1/n] and used continuity to take lebesgue measure ... etc.
How would you do this with open and closed sets?
Finally for showing l isn't sigma finite on 5, is it enough to say that whatever increasing sequence you take in C_half, using continuity of the lebesgue, when you do the lebesgue measure will always get ∞ + b where b is in the reals which is clearly never going to be finite?


Thanks a lot for any help, I'm really stuck on these and theyre really important in my module!

 

[mighty2000]

Dear forum post author,

Thank you for your question. It seems like you have made good progress in proving that 1-4 generate the borel sigma algebra and that lebesgue is sigma finite on 1 and 2. For 3 and 4, you can use a similar approach as you did for 2. Let A_n = (-n,n) for n ∈ N. Then, A_n is an increasing sequence of open sets that cover the reals, and you can use continuity of lebesgue measure to show that l(A_n) < ∞ for all n.

As for showing that C_half generates the borel sigma algebra, you can use a similar approach as you did for 1-4. Let A be an open set in the borel sigma algebra. Since A is open, it can be written as a union of open intervals, i.e. A = ∪_i (a_i, b_i). Now, for each i, there exists a rational number r_i such that a_i < r_i < b_i. Let A_i = (-∞, r_i]. Then, A_i is an increasing sequence of sets in C_half and ∪_i A_i = (-∞, b_i) ⊆ A. Thus, A is a countable union of sets in C_half, and therefore A ∈ σ(C_half). This shows that σ(C) = σ(C_half), and therefore C_half also generates the borel sigma algebra.

Finally, for showing that lebesgue is not sigma finite on C_half, your reasoning is correct. Any increasing sequence of sets in C_half will have infinite lebesgue measure, since it will always contain a set of the form (-∞, b] for some b ∈ R, which has infinite measure.

I hope this helps you in completing your proof. Good luck with your module!