Definition of “measurable” in context of random variables

For a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, we say $X: \Omega \rightarrow \mathbb{R}$ is a random variable if $X^{-1}(B) \in \mathcal{F} ~\forall B \in \mathcal{B}$, where $\mathcal{B}$ is the Borel $\sigma$-field. In other words, $X$ is a random variable if $X$ is measurable.

The intuition I've gathered so far is that since we can only talk about measurable subsets of $\Omega$, we want the preimage of any set in the image of X to be measurable. However, my question is, why are we only concerned with preimages of Borel sets, rather than all subsets of $\mathbb{R}$?

(For context, my experience with measure theory is only in the view of probability theory)

• Measurability is always related to two sigma-algebras, one on the source set and the other on the target set. There is no reason to chose the power set as the latter (and actually one rarely does so). – Did Oct 31 '16 at 21:37
• The equivalence of the two definitions seems to imply that the Borel $\sigma$-field is the largest collection of measurable subsets of $\mathbb{R}$, but I feel like that's not the case. – dkv Oct 31 '16 at 21:38
• "The equivalence of the two definitions" What? If "the two definitions" are what I think, then they are not equivalent (and one of them is never used). – Did Oct 31 '16 at 21:43
• $X^{-1}(B) \in \mathcal{F} ~\forall B \in \mathcal{B}$ vs "$X$ is measurable." Do these not mean the same thing? – dkv Oct 31 '16 at 21:47
• You might want to check this – Ran Wang Feb 23 '17 at 20:23

In fact, it is enough to be concerned about sets $\{X > a\}$ with $a$ real.
For $X$ to be considered a "random variable" we need to be able to tell whether or not it is ${} > a$ for any given real number $a$.
You can try to work with "all subsets of $\mathbb R$", but you won't get very far. Why not use only minimal assumptions? We do not want to use the largest possible collection of sets; that is, we do not want to make it most difficult to be a "random variable". Instead we want to use the smallest possible collection of sets; that is, we want to make it very easy to be a "random variable".
• Right, but that's exactly equivalent to the Borel-set definition above since $(a,\infty)$ is a generating set for $\mathcal{B}$. – dkv Oct 31 '16 at 21:55
• Why only a constant? What about {a<X<b} or {a<X<b and c<X<d} or {c<X<d and X=4}? Sure, those can be turned into comparisons to a constant, but I can keep going and in the most general case ask about {X $\in$ S} for any S $\subset \mathbb{R}$? – dkv Oct 31 '16 at 22:11