Currently in my undergraduate courses I am being taught how to set up various machinery using slick, short proofs and then how to apply that machinery. What I am not being taught, largely, is what came before these slick, short proofs. What did mathematicians do before so-and-so proved such-and-such lemma? Where, in other words, are the tedious, long proofs that we can look to as examples of the horrible mess we are escaping? What insights helped mathematicians escape those messes?

Right now I am particularly interested in examples from measure theory. What did people do before, for example, Dynkin's lemma or Caratheodory's extension theorem? Or were these tools available from near the start?

An answer should include both some indication of how tedious and long the old approach was and how much slicker and shorter the modern approach is. Ideally, it should also discuss how the transition between the two happened.

(If you prefer the old approach to the modern approach, for example for pedagogical reasons, that would also be interesting to hear about.)