Measure Theoretic Definition of a Random Variable I am struggling a little with the definition of a RV: Let $(\Omega,F),(\Omega',F')$ be two event spaces. Then every mapping: $X:\Omega \to\Omega'$ is a RV provided $X^{-1}A' \in F,~~~ \forall A' \in F'$.

So my understanding of this: Let $T:X \to Y$ and $S\subseteq Y$. $T^{-1}(S)$ is the preimage of S under T. Now, $T(T^{-1}(S)\subseteq S $ and not necessarily:$T(T^{-1}(S)= S $. However, in the case of a random variable, we must have equality. 

Is this the correct way to think about it?
The underlying idea ( in very practical terms) is simple. Suppose you are a statistician working with some survey. Lets suppose the survey has some questions about age, but only ask the respondent to identify his age in some given intervals, like $[0,18), [18, 25), [25,34), \dots $. Lets forget the other questions. This questionnaire defines an "event space", your $(\Omega,F)$. The sigma algebra $F$ codifies all information which can be obtained from the questionnaire, so, for the age question (and for now we ignore all other questions), it will contain the interval $[18,25)$ but not other intervals like $[20,30)$, since from the information obtained by the questionnaire we cannot answer question like: do the respondents age belong to this interval or not? More generally, a set is an event (belongs to $F$) if and only if we can decide if a sample point belongs to that set or not. 

Now, let us define random variables with values in the second event space, $(\Omega', F')$. As an example, take this to be the real line with the usual (Borel) sigma-algebra. Then, an (un-interesting) function which is not a random variable is $f: $"respondents age is a prime number", coding this as 1 if age is prime, 0 else. No, $f^{-1}(1)$ do not belong to $F$, so $f$ is not a random variable. The reason is simple, we cannot decide from the information in the questionnaire if the respondents age is prime or not!  No you can make more interesting examples yourself.  

I saw this kind of introduction first in the very good book by Peter Whittle "Probability via expectation" (Springer).
The underlying idea ( in very practical terms) is simple. Suppose you are a statistician working with some survey. Lets suppose the survey has some questions about age, but only ask the respondent to identify his age in some given intervals, like $[0,18), [18, 25), [25,34), \dots $. Lets forget the other questions. This questionnaire defines an "event space", your $(\Omega,F)$. The sigma algebra $F$ codifies all information which can be obtained from the questionnaire, so, for the age question (and for now we ignore all other questions), it will contain the interval $[18,25)$ but not other intervals like $[20,30)$, since from the information obtained by the questionnaire we cannot answer question like: do the respondents age belong to this interval or not? More generally, a set is an event (belongs to $F$) if and only if we can decide if a sample point belongs to that set or not. 

Now, let us define random variables with values in the second event space, $(\Omega', F')$. As an example, take this to be the real line with the usual (Borel) sigma-algebra. Then, an (un-interesting) function which is not a random variable is $f: $"respondents age is a prime number", coding this as 1 if age is prime, 0 else. No, $f^{-1}(1)$ do not belong to $F$, so $f$ is not a random variable. The reason is simple, we cannot decide from the information in the questionnaire if the respondents age is prime or not!  No you can make more interesting examples yourself.  

I saw this kind of introduction first in the very good book by Peter Whittle "Probability via expectation" (Springer).