Borel sets are defined because we want a measurable set of sets, and we want them to have nice topological properties.
.... Uhm.... but when do we actually take the size of Borel sets? When are they ever measured?
In Probability Theory, we measure $X \in A$, where $A$ is a Borel set. But we are not measuring a Borel set, since $X \in A$ is not a Borel set. It lies in the abstract sigma-algebra $\mathcal{F}$.
So, when do we actually measure a Borel set?