The OP
here says "After all, to define the Lesbesgue integral you could take the Caratheodary route or the representation theory route - both fiddly - but at least you know why you were putting the effort in." -- What is this representation theory route towards Lebesgue integral? I wouldn't understand surely; if you can just point me to some relevant keywords/wikipedia page/lecture notes, I'll be happy. =)