Lebesgue integral

Imagine counting the money in a pile of coins. Riemann's method walks left to right along the table and adds up whatever it meets, slice by slice along the input. Lebesgue's method instead sorts the coins by denomination first — all the pennies together, all the dimes together — and then adds value times count. The Lebesgue integral slices the range of the function (the output values), not the domain, and weighs each value by the measure of the set where the function takes it.

Concretely, for a non-negative measurable function f on a measure space, one approximates f from below by simple functions (finite combinations of indicators of measurable sets), integrates those exactly as value-times-measure sums, and takes the supremum. For a general real or complex f one splits into positive and negative (or real and imaginary) parts and combines, provided the pieces have finite integral. The result is written as the integral of f dμ over the space, or over a measurable set.

Because it measures sets rather than relying on the geometry of intervals, the Lebesgue integral handles wildly discontinuous functions, passes to limits under mild hypotheses, and works on abstract spaces. It is the foundation of the L^p spaces, of probability theory, and of most of modern analysis. Honest caveat: it does not magically make every function integrable — an improper Riemann integral that converges only conditionally, like the integral of (sin x)/x over the positive reals, is not Lebesgue integrable, because the absolute value has infinite integral.