Definition:Measurable Set

Measurable Sets of an Arbitrary Outer Measure
Given an outer measure $$\mu^*\ $$ on a set $$X\ $$, a subset $$S\subseteq X$$ is called $$\mu^*\ $$-measurable if it satisfies the so-called Carathéodory condition:


 * $$\mu^*(A) = \mu^*(A\cap S) + \mu^*(A - S)$$

for every $$A\in\mathcal P(X)$$. The collection of $$\mu^*\ $$-measurable sets is denoted $$\mathfrak M(\mu^*)$$ and is a $\sigma$-algebra over $X$.

Measurable Subsets of the Reals
A subset $$E \ $$ of the reals is said to be Lebesgue measurable, frequently just measurable, if for every set $$A \in \mathbb{R} \ $$,

$$m^*A = m^*(A \cap E) + m^*(A \cap \mathcal{C} \left ({E}\right))$$

where $$m^* \ $$ is defined as described in the definition of Lebesgue measure and $$\mathcal{C} \left ({E}\right)$$ is the complement of $$E \ $$ in $$\mathbb{R}$$.

The set of all measurable sets of $$\mathbb{R}$$ is frequently denoted $$\mathfrak {M}_\mathbb{R}$$ or just $$\mathfrak {M}$$.

There are sets in $\mathcal{P} \left({\mathbb{R}}\right)$ which are not in $\mathfrak {M} \ $.

Measurable Subsets of $$\mathbb{R}^n \ $$
A subset $$E \ $$ of $$\mathbb{R}^n \ $$ is said to be Lebesgue measurable, frequently just measurable, if for every set $$A \in \mathbb{R}^n \ $$,

$$m^*A = m^*(A \cap E) + m^*(A \cap \mathcal{C} \left ({E}\right))$$

where $$m^* \ $$ is defined as

$$m^*(E) = \inf_{\left\{{I_k}\right\} :E \subseteq \cup I_k} \sum v (I_k) \ $$

where $$\left\{{I_k}\right\} \ $$ are a sequence of sets satisfying

$$I_k = [a_1,b_1] \times \dots \times [a_n,b_n] $$

In the definition, infimum ranges over all such sets $$\left\{{I_n}\right\} \ $$, and $$v(I_n) \ $$ is the "volume" $$\Pi_{i=1}^n |b_i-a_i| \ $$, and $$\mathcal{C} \left ({E}\right)$$ is the complement of $$E \ $$ in $$\mathbb{R}^n$$.

The set of all measurable sets of $$\mathbb{R}$$ is frequently denoted $$\mathfrak {M}_{\mathbb{R}^n}$$.