1 Introduction to Probability

• ${\displaystyle \mathrm{\Omega }}$ is some space, representing the set of all outcomes.
• ${\displaystyle \mathcal{F}}$ is a ${\displaystyle \sigma -}$ algebra (${\displaystyle \sigma -}$ field) on ${\displaystyle \mathrm{\Omega }}$, representing a set of subsets of ${\displaystyle \mathrm{\Omega }}$ satisfying certain properties.
• ${\displaystyle \mathbb{P}}$ is probability measure.

Given a set ${\displaystyle \mathcal{S}}$, ${\displaystyle \mathcal{F}\subset \mathcal{P}(\mathcal{S})}$ is called a ${\displaystyle \sigma -}$ algebra on ${\displaystyle \mathcal{S}}$ if

• ${\displaystyle \varnothing ,\mathcal{S}\in \mathcal{F}}$;
• ${\displaystyle A\in \mathcal{F}\Rightarrow A^{\mathrm{c}}\in \mathcal{F}}$;
• (Closed under countable union) ${\displaystyle A_i\in \mathcal{F},i\in \mathbb{N}\Rightarrow \bigcup _{i=1}^{\mathrm{\infty }}{A}_i\in \mathcal{F}}$.

${\displaystyle \mathcal{F}=\{\varnothing ,\mathcal{S}\}}$; ${\displaystyle \mathcal{F}=\mathcal{P}(\mathcal{S})}$. They are respectively smallest/largest ${\displaystyle \sigma -}$ algebra.

${\displaystyle (\mathcal{S},\mathcal{F})}$ (${\displaystyle \mathcal{F}}$ is a ${\displaystyle \sigma -}$ algebra on ${\displaystyle \mathcal{S}}$) is a measurable space.
A non-negative set function ${\displaystyle \mu :\mathcal{F}\to [0,\mathrm{\infty }]}$ is called a measure on ${\displaystyle (\mathcal{S},\mathcal{F})}$, if

• ${\displaystyle \mu (\varnothing )=0}$;
• ${\displaystyle \mathrm{\forall }{A}_i\in \mathcal{F},i\in \mathbb{N}}$ s.t. ${\displaystyle A_i\cap A_j=\varnothing }$. If ${\displaystyle i\ne j}$, $$\mu \left(\bigcup _{i=1}^{\mathrm{\infty }}{A}_i\right)=\sum _{i=1}^{\mathrm{\infty }}\mu (A_i).$$

Consider a measure ${\displaystyle \lambda }$ on ${\displaystyle (\mathbb{R},\mathcal{P}(\mathbb{R}))}$ satisfying

• ${\displaystyle \lambda ([a,b])=b-a,b>a}$;
• ${\displaystyle \lambda (x+A)=\lambda (A),x\in \mathbb{R},A\in \mathcal{P}(\mathbb{R})}$.

Then ${\displaystyle \mathrm{\exists }V\in \mathcal{P}(\mathbb{R})}$ for which ${\displaystyle \lambda (V)}$ cannot be defined consistently, like Vitally set. I.e. not Lebesgue measurable.
For example, consider a unit ball ${\displaystyle B\subset {\mathbb{R}}^3}$ and drop a point ${\displaystyle x}$ u.a.r on ${\displaystyle B}$. For any subset ${\displaystyle A\subset B}$, we can't define ${\displaystyle \mathbb{P}(x\in A)=\frac{\mathrm{Volume}(A)}{\frac{4}{3}\pi }}$. By Banach-Tarski Theorem, some ${\displaystyle A\in \mathcal{P}(B)}$ are not Lebesgue measurable.