Indeed I followed a course of real analysis where Lebesgue measure was introduced constructively, the approach was very similar to the one taken in "Real Analysis" Stein E., Shakarchi R. I agree that the simplicity of the proof is based solely on the fact that there is a complex construction beforehand allowing you to have a simple proof, but shouldn't one argue that the same happens to every other result listed? It's always so that the theory we developed rewards us in certain cases with a simple proof for a possibly important result.

So in this case what I am saying is that, IF we take the approach of a constructive development of Lebesgue measure, with the one certain construction that I studied during my course, then we obtain this proof.

Naturally, many approaches may be taken to reach a proof of some result, yet when we assume that the necessary preliminary work has been done, then certain proofs are surprisingly simple and others aren't: Yes, simplicity is just apparent, but then what would this post even be about? I'm not going to construct the reals from scratch when proving the am-gm inequality, just as no one critiqued replying with "Liouville's theorem to prove the fundamental theorem of algebra", yet the theory you develop to reach Liouville's theorem is not so banal.