Lebesgue Measure on Rn
Our goal is to construct a notion of the volume, or Lebesgue measure, of rather
general subsets of Rn that reduces to the usual volume of elementary geometrical
sets such as cubes or rectangles.
If L(Rn ) denotes the col
Measures are a generalization of volume; the fundamental example is Lebesgue
measure on Rn , which we discuss in detail in the next Chapter. Moreover, as
formalized by Kolmogorov (1933), measure theory provides the foundation of probabi
Measurable functions in measure theory are analogous to continuous functions
in topology. A continuous function pulls back open sets to open sets, while a
measurable function pulls back measurable sets to measurable sets.
Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R2 as the
product of Lebesgue measures on R. The integral of a measurable funct
The generalization from elementary calculus of dierentiation in measure theory
is less obvious than that of integration, and the methods of treating it are somewhat
Consider the fundamental theorem of calculus (FTC) for s