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

Measure Thoery: Chp 4. Signed and Complex Measures

In this chapter we study signed and complex measures, which are defined to be the countably additive functions from a -algebra to or to that have value on the empty set.

Signed and Complex Measures

Let be a measurable space, and let be a function on with values in . If is countably additive and satisfies , then it is a signed measure.

A complex measure on is a function from to that satisfies and is countably additive.

Theorem 4.1.5 (Hahn Decomposition Theorem). Let be a measurable space, and let be a signed measure on . Then there are disjoint subsets and of such that is a positive set for , is a negative set for , and .

Corollary 4.1.6 (Jordan Decomposition Theorem). Every signed measure is the difference of two positive measures, at least one of which is finite.

Absolute Continuity

Let be a measurable space, and let and be positive measures on . Then is absolutely continuous with respect to if each set that belongs to and satisfies also satisfies . One sometimes writes to indicate that is absolutely continuous with respect to .

Theorem 4.2.2 (Radon–Nikodym Theorem). Let be a measurable space, and let and be -finite positive measures on . If is absolutely continuous with respect to , then there is an -measurable function such that holds for each A in . The function is unique up to -almost everywhere equality.

Singularity

Let be a measurable space. A positive measure on is concentrated on the -measurable set if .

Suppose that and are positive, signed, or complex measures on . Then and are mutually singular if there is an -measurable set such that is concentrated on and is concentrated on . ()

Theorem 4.3.2 (Lebesgue Decomposition Theorem). Let be a measurable space, let be a positive measure on , and let be a finite signed, complex, or -finite positive measure on . Then there are unique finite signed, complex, or positive measures and on such that - is absolutely continuous with respect to , - is singular with respect to , and -

The decomposition is called the Lebesgue decomposition of , while and are called the absolutely continuous and singular parts of .

Functions of Finite Variation

The Duals of the Spaces

Theorem 4.5.1. Let be a measure space, let satisfy , and let be defined by . If and is -finite, or if and is arbitrary, then the operator defined above is an isometric isomorphism of onto .

References

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