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 $\displaystyle X$? 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 $\displaystyle X$ is some set, and $\displaystyle \sigma(X)$ is the smallest $\displaystyle \sigma$-algebra containing $\displaystyle X$. You are looking for a map $\displaystyle f:X \to \mathbb{R}$ such that there exists a set $\displaystyle A \subseteq \mathbb{R}$ such that $\displaystyle A$ is measurable, $\displaystyle f^{-1}(A)$ is not in $\displaystyle \sigma(X)$. 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 $\displaystyle R \subseteq \mathbb{R}\times \mathbb{R}$ by $\displaystyle aRb$ if and only if $\displaystyle b-a \in \mathbb{Q}$. Show that $\displaystyle R$ is an equivalence relation. Then, partition $\displaystyle \mathbb{R}$ by $\displaystyle R$. Choose a representative from each partition where $\displaystyle x_R$ is the representative chosen from the part containing $\displaystyle x$. Define a function $\displaystyle f:\mathbb{R} \to \mathbb{R}$ by $\displaystyle f(x) = x_R$. That should do it.