Mathematics
Measure Theory and the Definition of the Lebesgue Integral
Quick fact
The Lebesgue integral can integrate functions that the Riemann integral cannot, such as the indicator function of the rational numbers on [0,1] (which has integral 0), and it makes interchange of limits and integration much easier.
Why this is interesting
You've learned to integrate by slicing vertically into thin rectangles. But what if you sliced horizontally instead? That simple shift unlocks a whole new world of integrals.