Measure Spaces

Σ-Algebras

Definition: σ-Algebra

Let $X$ be a set.

A σ-algebra is a collection $\Sigma$ of subsets of $X$ with the following properties:

• The empty set $\varnothing$ and $X$ itself are in $\Sigma$, i.e. $\varnothing ,X\in \Sigma$;
• If a subset $S$ is in $\Sigma$, then so is its complement $X\setminus S$, i.e. $S\in \Sigma ⟹X\setminus S\in \Sigma$;
• If $\mathcal{S}$ is a countable subcollection of $\Sigma$, then its union $\bigcup \mathcal{S}$ is also in $\Sigma$.

Theorem: Countable Intersections of σ-Algebras

Let $X$ be a set and let $\Sigma$ be a σ-algebra on $X$.

If $\mathcal{S}$ is a countable subcollection of $\Sigma$, then its intersection $\bigcap \mathcal{S}$ is also in $\Sigma$.

PROOF

TODO

Definition: Measurable Space

A measurable space $(X,\Sigma )$ is a set $X$ equipped with a σ-algebra $\Sigma$ on it.

Definition: Measurable Sets

The elements of $\Sigma$ are called measurable sets.

Measures

Definition: Measure

Let $(X,\Sigma )$ be a measurable space.

A measure on $(X,\Sigma )$ is a function $\mu :\Sigma \to {\mathbb{R}}_{\ge 0}\cup \{\infty \}$ from $\Sigma$ to the non-negative extended real numbers with the following properties:

• $\mu (\varnothing )=0$;
• If $\{S_i{\}}_{i=1}^{\infty }$ is a countable subcollection of $\Sigma$ consisting of pairwise disjoint sets, then

$$\mu \left(\bigcup _{i=1}^{\infty }{S}_i\right)=\sum _{i=1}^{\infty }\mu (S_i)$$

NOTATION

We usually write ${\mathbb{R}}_{\ge 0}\cup \{\infty \}$ as $[0,\infty ]$.

Definition: Measure Space

A measure space $(X,\Sigma ,\mu )$ is a measurable space $(X,\Sigma )$ equipped with a measure $\mu$ on it.

Definition: Null Set

Let $(X,\Sigma ,\mu )$ be a measure space.

A subset $S\in \Sigma$ is a null set if its measure is zero.

$$\mu (S)=0$$

Definition: Complete Measure Space

A measure space $(X,\Sigma ,\mu )$ is complete if every subset of a null set is itself a null set.

$$S\subseteq N\in \Sigma \text{ and }\mu (N)=0⟹\mu (S)=0$$