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Mathematics

Riemann vs. Lebesgue Integral

Quick fact

The Lebesgue integral can integrate functions that the Riemann integral cannot, and it makes the limit theorems (like the monotone convergence theorem) work more cleanly, which is why it's the standard in modern analysis and probability.

Why this is interesting

You've probably computed areas under curves with the Riemann integral, but when functions get wild, it fails. What if there were a different way to integrate that could handle almost any function you can imagine?