Does every event map to a Borel set?

Question

Given a (real-valued) random variable $X$ in the probability space $(\Omega,\mathcal{F},P)$. Let $\mathcal{B}$ be the Borel sigma algebra on $\mathbb{R}$.

We know that given any $B \in \mathcal{B}$, $X^{-1}(B) \in F$. But is $X(A) \in \mathcal{B}$ for any $A \in F$?

Answer 1

No. Take your favorite example of a non-Borel set $S\subseteq \mathbb{R}$. Consider the probability space $(S,\mathcal{P}(S),P)$, where $P$ is any probability measure on $(S,\mathcal{P}(S))$ (it exists, take $P=\delta_x$ for some fixed $x\in S$). Then the function $f:S\to \mathbb{R}$ defined by $f(s)=s$ is measurable but $f(S)\notin\mathcal{B}$.