Lebesgue Measure

• categories: Measure Theory, Definition

The Lebesgue measure is a way to assign a “size” or “volume” to subsets of ${\mathbb{R}}^n$ that generalizes the intuitive concept of length, area, and volume. It extends beyond simple geometric shapes to more complex sets, making it a foundational tool in measure theory and integration.

Key Properties:

• Translation invariance:
  • If $A\subset {\mathbb{R}}^n$ has Lebesgue measure $m(A)$, then for any vector $v\in {\mathbb{R}}^n$, the translated set $A+v$ has the same measure: $m(A+v)=m(A)$
• Countable additivity:
  • If $A={\bigcup }_{i=1}^{\infty }{A}_i$ is a countable union of disjoint sets $A_i$, then $m(A)={\sum }_{i=1}^{\infty }m(A_i)$
• Null sets:
  • A set $A\subset {\mathbb{R}}^n$ has Lebesgue measure zero if for every $ϵ>0$, there exists a countable cover of $A$ by open intervals whose total length is less than $ϵ$

Formal Definition:

For an interval $I=[a_1,b_1]\times [a_2,b_2]\times \cdots \times [a_n,b_n]\subset {\mathbb{R}}^n$, the Lebesgue measure of $I$ is defined as:

$$m(I)=(b_1-a_1)(b_2-a_2)\cdots (b_n-a_n).$$

The Lebesgue measure of more general sets is defined through the concept of Outer Measure, which approximates sets using intervals, and is extended via Carathéodory’s criterion.

Intuition:

Lebesgue measure generalizes the concept of length (1D), area (2D), and volume (3D) to arbitrary sets in ${\mathbb{R}}^n$, allowing for a rigorous treatment of integration and convergence in analysis. It overcomes limitations of the Riemann integral by handling more irregular sets and functions.