# 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

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

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