Yesterday, I reviewed some principles behind the integration theory by Lebesgue.
In a few words: instead of slicing the domain of a function in small equal parts, Lebesgue slices the range of the function. This produces parts that are not of the same size (like the dx in Riemann) and makes possible to integrate functions that are not Riemann-integrable (for example for discontinuity).
Using this approach, the integral of a function should be the sum the elementary area contained in the thin horizontal strip between y = t and y = t − dt. This elementary area is just:
If we let 
then we can define the Lebesgue integral of a function 
as:
, where the integral on the right is an ordinary improper Riemann integral.
In order to do this, Lebesgue needs a measure to compute how big is a set. This measure is the same as area in 2D and as volume in 3D but generalises in multiple dimensions.
In case a function can be integrated both with Riemann and Lebesgue integral, then the value of the two integrals is the same.
Charles Pugh, in his book Real mathematical analysis, tells a funny story about counting money on a table in the Riemann and in the Lebesgue way. The first would collect all pieces and would sum them up. The second, instead, would organise first the money in groups of similar value. It’s a useful metaphor.
I think that the following image from Wikipedia is indeed meaningful…
