Measure Theory

σ-Algebras

A σ-algebra on a set $X$ is a collection $\mathcal{F}\subseteq \mathcal{P}(X)$ satisfying:

1. $X\in \mathcal{F}$
2. If $A\in \mathcal{F}$, then $A^c\in \mathcal{F}$
3. If $A_1,A_2,\dots \in \mathcal{F}$, then ${\bigcup }_{n=1}^{\infty }{A}_n\in \mathcal{F}$

Measures

A measure on $(X,\mathcal{F})$ is a function $\mu :\mathcal{F}\to [0,\infty ]$ such that:

1. $\mu (\varnothing )=0$
2. Countable additivity: For disjoint $\{A_n\}$, $\mu \left({\bigcup }_{n=1}^{\infty }{A}_n\right)={\sum }_{n=1}^{\infty }\mu (A_n)$

Lebesgue Measure

The Lebesgue measure $\lambda$ on ${\mathbb{R}}^n$ extends the notion of length/area/volume to a much larger class of sets than intervals and rectangles.