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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Sigma Albegra

**Physics Forums | Science Articles, Homework Help, Discussion**