Real valued Random Varaible - Please chech my definition

I am trying to understand Random Variables.

Please do let me know if my definiton is correct - or I have got things mixed up.

1. Let $\displaystyle \Omega$ be the sample space i.e. set of all possible outcomes of an experiment; This can be countable or uncountable.

2. $\displaystyle \omega$ be the sample point i.e. a specific outcome of the experiment

A real valued random variable is a

Function, $\displaystyle X:\Omega \rightarrow \mathbb{R}$

And that is it !!

Informally for every $\displaystyle \omega \in \Omega$, there is a unique $\displaystyle X(\omega) \in \mathbb{R} $

Is this definition correct / exact ?

Reason of my confusion is that some books define random variable as "a funciton which maps event in the given sample space to a real number." - I think this definition is wrong. Any comments?

I am trying to understand two concepts

1. E(X) - Expected value of X

2. E(X|A) - Conditional expectaion under event A

I think in 2 above, we have not changed the way X was defined on the sample space, only thing that changed is probabiltiy measure - Is this understanding correct?

Please help - Any references would be welcome