This is not about measure theory or Dynkin's lemma or Caratheodory's extension theorem, but it is hard for me to resist sharing one of my favorite examples of improving proofs with modern machinery: the Intermediate Value Theorem. This theorem is so intuitively obvious, but the proof using classical analysis involves taking a supremum of the set of $a \leq x \leq b$ such that $f(x) \leq y$ (where $y$ is the desired output) and then showing using continuity that this supremum $c$ satisfies $f(c) = y$. There are lots of $\delta$'s and $\epsilon$'s and the proof feels uninspiring and technical at best.

Enter topology. The proof that the image of a connected set is connected for a continuous function is simple and intuitive, as is the notion of a connected set. Once this is established, the Intermediate Value Theorem is essentially just the statement that an interval is a connected set, so the image must be connected. This proof captures, in my opinion, the intuition of the Intermediate Value Theorem in a precise way.