There is this definition of random variable:
Let $(\Omega,\mathcal{F}), (\Omega',\mathcal{F}')$ be two event spaces. We say that a function $X:\Omega\to\Omega'$ is a random variable from $(\Omega,\mathcal{F})$ to $(\Omega',\mathcal{F}')$ if $\forall$ $A'\in\mathcal{F}'$, we have $X^{-1}(A)\in\mathcal{F}$.
My question is why do we have to use preimage in the definition? What if we change it to $\forall$ $A\in\mathcal{F}$ then $X(A)\in\mathcal{F}'$?
Could anyone please provide some explanations and if possible some counterexamples on what would go wrong if we don't use the preimage in the definition?
Thanks.
Let me complement on the answer given by drhab.
Indeed, one would not be able in general to compute the measure of the set $X^{-1}A$. However, there is really no problem to introduce another notion of "random variable" by requiring that the image of a measurable set is a measurable set and proceed from there. Something similar (a bit peculiar though, from the point of view of dynamics) is done from time to time in the topological category, with the notion of a "proper map".
On the other hand, there would be serious problems in order to consider stationary probability measures since you would only be able to consider random variables that were invertible mod zero. From my point of view, this is the main reason why one should prefer the usual definition.