Absolute Continuity

Signed Measures and Total Variation

Signed Measure

A signed measure on a measurable space is a function such that and for any countable collection of disjoint sets in .

e.g. Suppose is a measure space, and is integrable. Then we can define a signed measure on by

Total Variation

Let be a signed measure on a measurable space . The total variation of , denoted by , is defined for any measurable set as:

Jordan Decomposition

Let be a signed measure on a measurable space . Then there exist two mutually singular finite non-negative measures and , called the positive variation and negative variation of , such that:

This decomposition is unique and is known as the Jordan decomposition of .

Absolute Continuity and Mutual Singularity

Support of a Measure

Given a measure space , a signed measure is said to be supported on a set , if for every , .

Mutually Singular Measures

Two signed measures and on a measurable space are said to be mutually singular, denoted , if there exists disjoint measurable sets such that is supported on and is supported on .

Absolute Continuity

Suppose is a measure and is a signed measure on a measurable space . We say that is absolutely continuous with respect to , denoted , if for every measurable set , if , then .

e.g. Suppose is a measure space, and is a measurable function. Then we can define another measure on by Then clearly that whenever , the integral , so . Moreover, if is -finite and then is also -finite.

Proposition

Suppose and are two measures on a measurable space . If and , then for all , there exits some such that implies for all measurable sets .

Proof We prove by contradiction. Suppose there exists some such that for all , there exists some measurable set with but . Consider the set . Then we have for all , so . However, we also have (ref. measure of decreasing intersection), which is a contradiction. Therefore, the proposition holds.

Radon-Nikodym Theorem

Suppose is a $\sigma$-finite measure and is a -finite signed measure on the same measurable space . There exits unique measures and , such that

• ,
• ,
• .

Moreover, there exits a unique integrable function such that for all measurable sets .

Remark

In the case that , , and . The first part of this theorem is known as the Lebesgue decomposition theorem, and the second part is known as the Radon-Nikodym theorem.

Proof