Lebesgue integral
Intuitive interpretation

Folland (1999) summarizes the difference between the Riemann and Lebesgue approaches thus: "to compute the Riemann integral of f, one partitions the domain into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f."

For the Riemann integral, the domain is partitioned into intervals, and bars are constructed to meet the height of the graph. The areas of these bars are added together, and this approximates the integral, in effect by summing areas of the form f (x) dx where f (x) is the height of a rectangle and dx is its width.

For the Lebesgue integral, the range is partitioned into intervals, and so the region under the graph is partitioned into horizontal "slabs" (which may not be connected sets). The area of a small horizontal "slab" under the graph of f, of height dy, is equal to the measure of the slab's width times dy: μ ({ x ∣ f (x) > y }) d y. {\displaystyle \mu \left(\{x\mid f(x)>y\}\right)\,dy.} The Lebesgue integral may then be defined by adding up the areas of these horizontal slabs. From this perspective, a key difference with the Riemann integral is that the "slabs" are no longer rectangular (cartesian products of two intervals), but instead are cartesian products of a measurable set with an interval.
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