# Very brief notes on measures: From σ-fields to Carathéodory’s Theorem

Definition 1. A ${\sigma}$-field ${\mathcal{F}}$ is a non-empty collection of subsets of the sample space ${\Omega}$ closed under the formation of complements and countable unions (or equivalently of countable intesections – note ${\bigcap_{i} A_i = (\bigcup_i A_i^c)^c}$). Hence ${\mathcal{F}}$ is a ${\sigma}$-field if

$1. {A^c \in \mathcal{F}}$ whenever ${A \in \mathcal{F}}$
$2. {\bigcup_{i=1}^{\infty} A_i \in \mathcal{F}}$ whenever ${A_i \in \mathcal{F}, n \geq 1}$

Definition 2. Set functions and measures. Let ${S}$ be a set and ${\Sigma_0}$ be an algebra on ${S}$, and let ${\mu_0}$ be a non-negative set function

$\displaystyle \mu_0: \Sigma_0 \rightarrow [0, \infty]$

• ${\mu_0}$ is additive if ${\mu_0 (\varnothing) =0}$ and, for ${F,G \in \Sigma_0}$,

$\displaystyle F \cap G = \varnothing \qquad \Rightarrow \qquad \mu_0(F \cup G ) = \mu_0(F) + \mu_0(G)$

• The map ${\mu_0}$ is called countably additive (or ${\sigma}$-additive) if ${\mu (\varnothing)=0}$ and whenever ${(F_n: n \in \mathbb{N})}$ is a sequence of disjoint sets in ${\Sigma_0}$ with union ${F = \cup F_n}$ in ${\Sigma_0}$, then

$\displaystyle \mu_0 (F) = \sum_{n}\mu_0 (F_n)$

• Let ${(S, \Sigma)}$ be a measurable space, so that ${\Sigma}$ is a ${\sigma}$-algebra on ${S}$.
• A map $\displaystyle \mu: \Sigma \rightarrow [0,\infty].$ is called a measure on ${(S, \Sigma)}$ if ${\mu}$ is countable additive. The triple ${(S, \Sigma, \mu)}$ is called a measure space.
• The measure ${\mu}$ is called finite if

$\displaystyle \mu(S) < \infty,$

and ${\sigma}$finite if

${\exists \{S_n\} \in \Sigma}$, (${n \in \mathbb{N}}$) s.th.$\displaystyle \mu(S_n)< \infty, \forall n \in \mathbb{N} \text{ and } \cup S_n = S.$

• Measure ${\mu}$ is called a probability measure if $\displaystyle \mu(S) = 1,$ and ${(S, \Sigma, \mu)}$ is then called a probability triple.
• An element ${F}$ of ${\Sigma}$ is called ${\mu}$-null if ${\mu(F)=0}$.
• A statement ${\mathcal{S}}$ about points ${s}$ of ${\mathcal{S}}$ is said to hold almost everywhere (a.s.) if

$\displaystyle F \equiv \{ s: \mathcal{S}(s) \text{ is false} \} \in \Sigma \text{ and } \mu(F)=0.$

Definition 3. A probability space is a triple ${(\Omega, \mathcal{F}, P )}$ where ${\Omega }$ is the sample space, ${\mathcal{F}}$ is the family of events (i.e. an event is an element of ${\mathcal{F}}$) which we assume to be a ${\sigma}$-field and ${P: \mathcal{F}\rightarrow [0,1]}$ is a function that assigns probabilities to events.

Definition 4. A non-empty class ${\mathcal{C}}$ of subsets of ${\Omega}$ is a monotone class if ${\lim C_n \in \mathcal{C}}$ for every monotone serquence ${C_n \in \mathcal{C}, n \geq 1}$.

Proposition 1. A ${\sigma}$-field is a monotone class while, conversely, a monotone algebra is a ${\sigma}$-field.

Corollary 1. Let ${2^X}$ be the power-set (the class of all subsets of ${\Omega}$) and ${T_{\Omega} = \{\varnothing ,\Omega\}}$.Then for any ${\sigma}$-field ${U_{\Omega}}$ of subsets of ${\Omega}$,

${ T_{\Omega} \subset U_{\Omega} \subset 2^X }$.

Definition 5. A real valued function ${X}$ defined on ${\Omega}$ is said to be a random variable if for every Borel set ${B \in \mathbb{R}}$ we have

$\displaystyle X^{-1} (B) = \{\omega: X(\omega) \in B\} \in \mathcal{F}$

A random variable is an element of ${m \mathcal{F}}$. Thus

$\displaystyle X: \Omega \rightarrow \mathbb{R}, \qquad X^{-1} : \mathcal{B} \rightarrow \mathcal{F}.$

Definition 6. A function ${X: \Omega \rightarrow S}$ is said to be a measurable map from ${(\Omega,\mathcal{F})}$ to ${(S, \mathcal{S})}$ if

$\displaystyle X^{-1}(B) \equiv \{ \omega: X(\omega) \in B \} \in \mathcal{F} \text{ } \forall B \in \mathcal{S}$

Theorem 1. If ${\{\omega: X(\omega) \in A\} \in \mathcal{F}}$ ${\forall A \in \mathcal{A}}$ and ${ \mathcal{A} \text{ generates } \mathcal{S}}$, then ${X}$ is measurable. A ${\sigma}$-field generated by a class ${\mathcal{C}}$ of subsets (denoted ${\sigma(\mathcal{C})}$) is the smallest ${\sigma}$-field on ${\Omega}$ that makes ${\mathcal{C} }$ a measurable map. That is ${ \mathcal{C} \subset \sigma(\mathcal{C}) \subset \mathcal{B'}}$ where ${\mathcal{B'} }$ is any other subset containing ${\mathcal{C}}$.

Proposition 2. If ${\{ \mathcal{F}_{i} \}_{i \in I} }$ is the collection of all ${\sigma}$-fields that contain ${\mathcal{C}}$ then

$\displaystyle \sigma(\mathcal{C}) = \bigcap_{i \in I} \mathcal{F}_i$

Definition 8. The Borel ${\sigma}$-field ${\mathcal{B}(\mathbb{R})}$ is the smallest ${\sigma}$-field containing all open intervals of ${\mathbb{R}}$. ${\mathcal{B}(\mathbb{R})}$ is generated by each of the following

• the collection of all closed subsets of ${\mathbb{R}}$;
• the collection of all subintervals of ${\mathbb{R}}$ of the form ${(-\infty, b]}$;
• the collection of all subintervals of ${\mathbb{R}}$ of the form ${(a, b]}$.

Theorem 2. If ${X: (\Omega, \mathcal{F}) \rightarrow (S, \mathcal{S})}$ and ${f: (S, \mathcal{S}) \rightarrow (T, \mathcal{T}) }$ are measurable maps, then ${f(X)}$ is a measurable map from ${(\Omega, \mathcal{F})}$ to ${(T, \mathcal{T}) }$.

Theorem 3. If ${X_1,...,X_n}$ are random variables and ${f:(\mathbb{R}^n, \mathcal{R}^n) \rightarrow (\mathbb{R}, \mathcal{R})}$ is measurable, then ${f(X_1,...,X_n)}$ is a random variable.

Theorem 4. If ${X_1, X_2...}$ are random variables then so are

$\displaystyle \inf_n X_n \qquad \sup_n X_n \qquad \limsup_n X_n \qquad \liminf_n X_n$

Definition 9. ${X_n}$ converges almost surely (a.s.) if ${P(\Omega_0) =1}$ where

$\displaystyle \Omega_0 \equiv \{\omega : \lim\limits_{n \rightarrow \infty} X_n \text{ exists } \} = \{\omega: \limsup_{n\rightarrow\infty} X_n - \liminf_{n\rightarrow\infty} X_n = 0 \}$

Definition 10. A ${\pi}$-system ${\mathcal{I}}$ on a set ${S}$ is a family of subsets of ${S}$ stable under finite intersection:

$\displaystyle I_1, I_2 \in \mathcal{I} \Rightarrow I_1 \cap I_2 \in \mathcal{I}.$

Lemma 1. Let ${\Sigma \equiv \sigma(\mathcal{I})}$. Supposed ${\mu_1}$ and ${\mu_2}$ are measures on ${(S,\Sigma)}$ s.th. ${\mu_1(S) = \mu_2(S) < \infty}$ and ${\mu_1=\mu_2}$ on ${\mathcal{I}}$. Then

$\displaystyle \mu_1 = \mu_2 \text{ on } \Sigma$

Corollary 1. If two probability measures agree on a ${\pi}$-system then they agree on the ${\sigma}$-algebra generated by that ${\pi}$-system. Example: ${\mathcal{B} = \sigma(\pi(\mathbb{R}))}$.

Definition 11. Dynkin Class/ ${\mathcal{D}}$ or ${\mathcal{L}}$ system. Let ${S}$ be a set, and let ${\mathcal{D}}$ be a collection of subsets of ${S}$. Then ${\mathcal{D}}$ is called a d-system (on ${S}$) if

1. ${S \in \mathcal{D}}$,
2. if ${A,B \in \mathcal{D}}$ and ${A \subseteq B}$ then ${B-A \in \mathcal{D}}$
3. if ${A_n \in \mathcal{D}}$ and ${A_n \uparrow A}$, then ${A \in \mathcal{D}.}$

Recall ${A_n \uparrow A}$ means ${A_n \subseteq A_{n+1}}$ ${\forall n}$ and ${\cup A_n =A}$.

Definition 12. ${d(\mathcal{C})}$: The intersection of all d-systems which contain the class ${\mathcal{C}}$ of subsets of ${S}$. Subsequently ${d(\mathcal{C})}$, is a d-system, the smallest d-system which contains ${\mathcal{C}}$. Furthermore

$\displaystyle d(\mathcal{C}) \subseteq \sigma(\mathcal{C})$
.

Lemma 2. Dynkin’s lemma. If ${\mathcal{I}}$ is a ${\pi}$-system, then

$\displaystyle d(\mathcal{I}) = \sigma(\mathcal{I}).$

A d-system which contains a ${\pi}$-system contains the ${\sigma}$-algebra generated by that ${\pi}$-system.

Proposition 3. A collection ${\Sigma}$ of subsets of ${S }$ is a ${\sigma}$-algebra iff it is both a ${\pi}$-system and a d-system.

Theorem 5. Carathéodory’s Theorem. Let ${S }$ be a set, let ${\Sigma_0}$ be an algebra on ${S}$, and let

$\displaystyle \Sigma \equiv \sigma( \Sigma_0)$

If ${\mu_0}$ is a countably additive map ${\mu_0: \Sigma_0 \rightarrow [0, \infty], \text{ then } \exists \mu \text{ on } (S, \Sigma)}$ s.th.

$\displaystyle \mu = \mu_0 \text{ on } \Sigma_0.$