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Gunnar_Zarncke comments on What math is essential to the art of rationality? - Less Wrong Discussion
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My main aha-moment in statistics occurred when I encountered the lebesgue integral. Integrals suddenly generalized a lot. Lebesgue also allows a lot more nifty but intuitive integral transformations. And of course it is needed for dealing cleanly with probability densities.
Causal networks despite needing tricky rules follow from the other points on my list (trees and probability measures)