Definition:Measurable Set

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} \ $.