Outer Measure

• categories: Measure Theory, Definition
• properties: Nonadditivity of Outer Measure for Disjoint Subsets

Definition:

The outer measure is a generalization of the concept of length or volume to possibly more complex sets. Given a set $A\subseteq {\mathbb{R}}^n$, the outer measure of $A$, denoted by $m^{\ast }(A)$, is defined as:

$$m^{\ast }(A)=\inf \left\{\sum _{i=1}^{\infty }l(I_i)\mid A\subseteq \bigcup _{i=1}^{\infty }{I}_i,I_i\text{ are intervals}\right\}$$

where $l(I_i)$ denotes the length of the interval $I_i$, and the union ${\bigcup }_{i=1}^{\infty }{I}_i$ is a countable covering of $A$ by intervals.

Properties:

1. Monotonicity: If $A\subseteq B$, then $m^{\ast }(A)\le m^{\ast }(B)$.
2. Countable Subadditivity: For any countable collection of sets $A_i$, $$m^{\ast }\left(\bigcup _{i=1}^{\infty }{A}_i\right)\le \sum _{i=1}^{\infty }{m}^{\ast }(A_i)$$
3. Translation Invariance: If $T_c(A)=A+c$ for some constant vector $c$, then $m^{\ast }(T_c(A))=m^{\ast }(A)$.

Intuition:

Outer measure tries to assign a “smallest possible” value to a set by covering it with intervals and summing up their lengths. It serves as a foundation for defining more rigorous measures, such as the Lebesgue measure.