Give an example for a mapping from the sample space to R that is not a valid random variable.
What is "the sample space"? Is it just an arbitrary sample space ? What do you mean an example for a mapping that is not a valid random variable? Valid in what sense?
From what I gather from your question, I assume you mean that is some set, and is the smallest -algebra containing . You are looking for a map such that there exists a set such that is measurable, is not in . Is that what you mean?
yes, what you assumed is absolutely correct. I want an example for this. I think if I assume a set that is not a field or that does not contain all subsets of Omega (Whole set), may be then I can prove that mapping of this set to R will not result in a random variable...
Define a relation by if and only if . Show that is an equivalence relation. Then, partition by . Choose a representative from each partition where is the representative chosen from the part containing . Define a function by . That should do it.