Lebesgue Measurable Set

• categories: Measure Theory, Definition

Lebesgue Measurable Set

A set $E\subset {\mathbb{R}}^n$ is called Lebesgue measurable if it satisfies one of the following equivalent definitions:

1. Carathéodory’s criterion:

2. Approximability by Open Sets:

A set $E\subset {\mathbb{R}}^n$ is Lebesgue measurable if for every $ϵ>0$, there exists an open set $O\supset E$ such that:

$$m^{\ast }(O\setminus E)<ϵ.$$

This means that $E$ can be approximated from above by open sets.

3. Approximability by $F_{\sigma }$ Sets:

A set $E\subset {\mathbb{R}}^n$ is Lebesgue measurable if there exists an $F_{\sigma }$ set $F\supset E$ (a countable union of closed sets) such that:

$$m^{\ast }(F\setminus E)<ϵ.$$

This implies that $E$ can be approximated from above by $F_{\sigma }$ sets.

4. Null Set Modulo Outer Measure:

A set $E\subset {\mathbb{R}}^n$ is Lebesgue measurable if there exists a $G_{\delta }$ set $G\subset {\mathbb{R}}^n$ (a countable intersection of open sets) such that:

$$m^{\ast }(E△G)=0,$$

where $E△G$ denotes the symmetric difference between $E$ and $G$. This indicates that $E$ differs from a $G_{\delta }$ set by a null set.

5. Intersection with Every Set:

A set $E\subset {\mathbb{R}}^n$ is Lebesgue measurable if for every $ϵ>0$, there exists a measurable set $M$ such that:

$$m^{\ast }(E△M)<ϵ.$$

This suggests that $E$ can be approximated by a Measurable Set up to an arbitrarily small error in measure.

6. Borel Set Approximation:

A set $E\subset {\mathbb{R}}^n$ is Lebesgue measurable if there exists a Borel sets $B\supset E$ and $A\subset E$ such that:

$$m(B\setminus E)=0.$$

And

$$m(E\setminus A)=0.$$

This implies that $E$ can be approximated from below and from above by a Borel set, differing only by a null set.