Lebesgue Measurable Sets

Overview


Lebesgue measurable sets are the subsets of {% \mathbb{R}^n %} that can be assigned a consistent measure. The concept is generalized as Measurable Sets

Definition


Let {% E \subset \mathbb{R}^n %}

then {% E %} is measurable if
{% m^*(A) = m^*(A\cap E) + m^*(A \backslash E) %}
for all {% A \subset \mathbb{R}^n %}

Almost Everywhere


A condition {% C(x) %} is said to hold almost everywhere if the set {% x %} such that {% C(x) %} is false has measure {% 0 %}.