Let S={1,2,3,4,5,6}, F=σ(A1,A2), ie., the σ-algebra generated by A1 and A1 (the smallest σ-algebra containing A1 and A2) with A1={1,2,3,4} and A2={3,4,5,6}. Please complete the following:(adsbygoogle = window.adsbygoogle || []).push({});

a. List all sets in F

b. Is the random variable X(w)= 2, w=1,2,3,4; X(w)=7, w=5,6 measurable w.r.t. F?

c. Give an example of a r.v. on S that is NOT measurable w.r.t. F.

What I have done so far:

a. F={ø,1,2,3,4,5,6}

b. The definition of a measurable function: If F is a σ-algebra over X and T is a σ-algebra over Y, then a function f:X->Y is F-measurable if the preimage of every T is in F.

The preimage of T ={1,2,3,4,5,6} which is in F.

X={1,2,3,4,5,6} which is a σ-algebra over F.

We can let T={2,7} which could be a σ-algebra over Y.

Thus, X(w) is F-measurable.

c. Y(w)= 0, w=1,2,3,4,5,6; Y(w)=1, w>=7

This is not F-measurable since not all the preimages are in F.

**I am not sure if I did this problem correctly. In addition, I don't really know how to explain part b and c nicely. This is the first time I have seen a σ-algebra in my life. It would be nice if someone could help me with this. Any help/suggestion is great appreciated.

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# Homework Help: Sigma Albegra

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