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Relation between the viewpoints

One can think of the Lebesgue integral either in terms of slabs or simple functions. Intuitively, the area under a simple function can be partitioned into slabs based on the (finite) collection of values in the range of a simple function (a real interval). Conversely, the (finite) collection of slabs in the undergraph of the function can be rearranged after a finite repartitioning to be the undergraph of a simple function.

The slabs viewpoint makes it easy to define the Lebesgue integral, in terms of basic calculus. Suppose that f {\displaystyle f} is a (Lebesgue measurable) function, taking non-negative values (possibly including + ∞ {\displaystyle +\infty }). Define the distribution function of f {\displaystyle f} as the "width of a slab", i.e., F (y) = μ { x | f (x) > y }. {\displaystyle F(y)=\mu \{x|f(x)>y\}.} Then F (y) {\displaystyle F(y)} is monotone decreasing and non-negative, and therefore has an (improper) Riemann integral over (0, ∞) {\displaystyle (0,\infty)}. The Lebesgue integral can then be defined by ∫ f d μ = ∫ 0 ∞ F (y) d y {\displaystyle \int f\,d\mu =\int _{0}^{\infty }F(y)\,dy} where the integral on the right is an ordinary improper Riemann integral, of a non-negative function (interpreted appropriately as + ∞ {\displaystyle +\infty } if F (y) = + ∞ {\displaystyle F(y)=+\infty } on a neighborhood of 0).

Most textbooks, however, emphasize the simple functions viewpoint, because it is then more straightforward to prove the basic theorems about the Lebesgue integral.