Fubini’s Theorem

Abstract

Linear Lebesgue measure is defined by covering sequences of intervals, and plane measure by covering sequences of rectangles. We shall now consider how these measures are related to each other. It is clear what kind of answer we should expect. In elementary calculus we learn to compute the area between the graphs of two functions f ≦ g by the formula

$$\int_a^b {|f(x) - g(x)|dx.}$$

. Thus the area is computed “by slicing.” The generalization of this formula, which expresses the measure of any plane measurable set A as the integral of the linear measure of its sections perpendicular to an axis, is called Fubini’s theorem. We shall not formulate the theorem in full generality, but confine attention to the case in which A is a nullset. Then the theorem asserts that almost all vertical (or horizontal) sections of A have measure zero.