Example ($n$ coin tosses)

• Sample Space: $\Omega=\{ (r_{1},\dots,r_{n}),r_{i}\in \{ 0,1 \}, i=1,\dots,n \}$
• Event Space: $\mathcal{F}=2^{\Omega}$
• Probability Function: $P(r_{1},\dots,r_{n})=\frac{1}{2^{n}},\forall(r_{1},\dots,r_{n})\in\Omega$
• Probability Measure: $\mathbb{P}(A)=\frac{|A|}{2^{n}}$

Example (Uniform probability distribution on $[0,1]$)

Sample Space: $\Omega=[0,1]$ Event Space: Lebesgue Measurable σ-algebra Probability Measure: Outer Measure or $\mathbb{P}=\mathbb{P}^{*}$

Example ($\infty$ coin tosses)

• Sample Space: $\Omega=\{ (r_{1},r_{2},\dots),r_{i}\in \{ 0,1 \}, \forall i\ge 1 \}$
• Event Space: For any $n\ge 1$ and $a_{1},\dots,a_{n}\in \{ 0,1 \}$ set $$A_{a_{1},\dots,a_{n}}=\{ (r_{1},r_{2},\dots)\in\Omega:r_{i}=a_{i},1\le i\le n \}$$and then let $$\mathcal{J}:=\{ A_{a_{1},\dots,a_{n}}:n\ge 1,a_{1},\dots,a_{n}\in \{ 0,1 \}\}\cup \{ \emptyset,\Omega \} .$$We want (at minimum) a σ-algebra $\mathcal{F}\supseteq\mathcal{J}$ with:
• Probability Measure: $$\mathbb{P}(A_{\underbrace{ a_{1},\dots,a_{n} }_{ \in \mathcal{F} }})=\frac{1}{2^{n}},\quad\forall n\ge 1,a_{1},\dots,a_{n}\in \{ 0,1 \}$$