Integral and the difference between: The Riemann-Darboux approach & The Lebesgue approach.
Riemann-Darboux’s integration (in blue) and Lebesgue integration (in red). To get some intuition about the different approaches to integration, let us imagine that it is desired to find a mountain’s volume (above sea level).
The Riemann-Darboux approach: Divide the base of the mountain into a grid of 1 meter squares. Measure the altitude of the mountain at the center of each square. The volume on a single grid square is approximately 1 m2 × (that square’s altitude), so the total volume is the sum of the altitudes.
The Lebesgue approach: Draw a contour map of the mountain, where adjacent contours are 1 meter of altitude apart. The volume of earth contained in a single contour is approximately 1 m × (that contour’s area), so the total volume is the sum of these areas.
Professor Gerald B.Folland summarizes the difference between the Riemann and Lebesgue approaches thus: “to compute the Riemann integral of f, one partitions the domain [a, b] into subintervals”, while in the Lebesgue integral, “one is in effect partitioning the range of f “
( Gerald B.Folland (1999). Real analysis: Modern techniques and their applications. Pure and Applied Mathematics (New York) (Second ed.). New York: John Wiley & Sons Inc. xvi+386. )
See more at Integral on Wikipedia - Riemann integral - Lebesgue integration
Images, I shared at Integral - Contour line - Riemann sum - Serpentine wall at the University Virginia.
Another intuitive way of thinking of Lebesgue integrals (as described by Paul Montel): “I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.”