Henri Lebesgue (1875–1941) was a French mathematician who looked at the ancient question "what does it mean to measure something?" and gave an answer so good it rebuilt the whole idea of the integral from the ground up. If you have ever added things up, Lebesgue found a smarter way to do it.
The old way to find an area was to slice it into thin vertical strips. Lebesgue's trick was to
slice horizontally instead — grouping together all the points where a function has a
similar height. To make that rigorous he first needed
Lebesgue himself gave a lovely picture of his idea. Imagine emptying your pockets to count your money. One way is to add each coin as it comes out, in the order you pull them: that's the old integral. The smarter way is to sort them into piles first — all the pennies together, all the ten-cent pieces together — then count each pile and add up. That's the Lebesgue integral: group by value, not by position. Simple in words, revolutionary in maths.