Lebesgue Measurability

Measurable Sets

Lebesgue Measurable Set

A set is said to be Lebesgue measurable if for all , there is an open , such that If is Lebesgue measurable, we write for the exterior measure of .

We will see that there are several equivalent ways to define a Lebesgue measurable set in the end.

Proposition

If , then is Lebesgue measurable. Such sets are called null sets.

Proof For any , we can choose open set with . Then .

e.g. , (standard middle-thirds) Cantor set are null sets in .

Proposition

Open sets in are Lebesgue measurable.

Proof Straightforward from the definition.

Proposition

A countable union of measurable sets is measurable.

Proof Suppose are measurable sets in . For all , choose open with . Then and we have

Proposition

Closed sets in are Lebesgue measurable.

Proof Suppose is closed. Without loss of generality, we can assume is compact.

Lemma

If is closed, is compact, and these sets are disjoint, then .

Proposition

The complement of a measurable set is measurable.

Corollary

A countable intersection of measurable sets is measurable.

Proof The complement of a countable union is a countable intersection.

Theorem

If are disjoint measurable sets, and , then

Proof

Corollary

For any two measurable sets and , we have

Proof We add the following two equations together:

Corollary

Suppose are measurable subsets of .

1. If , then .
2. If and for some , then

Where if is a countable collection of subsets of that increases to in the sense that for all , and , then we write . Similarly, if is a countable collection of subsets of that decreases to in the sense that for all , and , then we write .

Theorem

Suppose is a measurable subset of . Then, for every :

1. There exists an open set with and .
2. There exists a closed set with and .
3. If is finite, there exists a compact set with and .
4. If is finite, there exists a finite union of closed cubes such that
   

Invariance Properties

Translation Invariance

is a measurable set iff for all , the set is measurable, and .

Proof This is easy to see by replacing with .

Dilation Invariance

Suppose , and denote by the set , then is measurable iff is measurable, and .

Proof This is easy to see by replacing with , and dilation of an open set remains open. Moreover, for every cube , we have , so .

Reflection Invariance

is measurable iff is measurable, and .

Proof This is easy to see by replacing with .

Sigma Algebra and Borel Sets

Lebesgue -Algebra

The Lebesgue -algebra on is the [[Measurable Spaces and Functions#^60a516|-algebra]] generated by all Lebesgue measurable sets.

Borel -Algebra

The Borel -algebra on , denoted as , is the smallest -algebra containing all open sets. The term “smallest” means that if is any -algebra that contains all open sets in , then necessarily . Elements of the Borel -algebra are called Borel sets.

Corollary

A subset of is measurable

1. if and only if differs from a by a set of measure zero, i.e., , and .
2. if and only if differs from an by a set of measure zero. i.e., , and .

Where a set is a countable intersection of open sets, and an set is a countable union of closed sets.

Here is a summary of the equivalence of the definitions of a Lebesgue measurable set:

Equivalences for Lebesgue Measurability

Suppose . Then the following are equivalent:

1. is Lebesgue measurable.
2. For each , there exists a closed set with .
3. There exist closed sets contained in such that
   
4. There exists a Borel set such that .
5. For each , there exists an open set such that .
6. There exist open sets containing such that
   
7. There exists a Borel set such that .

Non Measurable Sets

We now explicitly construct a non-measurable set.

Proposition

Consider , we identify a set of representatives of each equivalence class in . Then is a non-measurable set.

Proof We shall prove by contradiction. Suppose is measurable.

Axiom of Choice

The construction of such a set relies on the axiom of choice, and the existence of non-measurable sets is independent of the standard axioms of set theory.