Notes on book "Measure Theory (2nd ed.) - Cohn Donald L."

Measure Thoery: Chp 6. Differentiation

In this chapter we look at two aspects of the relationship between differentiation and integration.

Change of Variable in

Let be the set of all by matrices with real entries, and let be a real-valued function on . - The function is multilinear if for each and each choice of the map is linear, - is alternating if holds whenever two of the columns of are equal, and - is a determinant if it is multilinear, is alternating, and satisfies

Differentiation of Measures

Theorem 6.2.1 (Vitali Covering Theorem). Let be an arbitrary nonempty subset of , and let be a Vitali covering of . Then there is a finite or infinite sequence of disjoint sets that belong to and are such that contains -almost every point in A.

Differentiation of Functions

References

[1] @book{cohn2013measure, title={Measure theory}, author={Cohn, Donald L}, year={2013}, publisher={Springer} }