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1.2 Probability Space
1.2.1 Probability Space
• Probability Space = $ \left\{ \mathcal{S},\mathcal{F}\left(\mathcal{S}\right),\mathcal{P}\right\} $
• $ \mathcal{S}\sim $ sample space
• $ \mathcal{F}\left(\mathcal{S}\right)\sim $ event space , collection of subsets of $ \mathcal{S} $ (including sample space itself)
• $ \mathcal{P}\sim $ maps $ \mathcal{F}\left(\mathcal{S}\right)\rightarrow\left[0,1\right] $
1.2.2 Event space
• Event space $ F\left(S\right) $ or $ F $ is a non-empty collection of subset of $ S $ satisfying the following three closure properties:
1. If $ A\in F\left(S\right) $ , then $ \bar{A}\in F\left(S\right) $ .
2. If for some finite $ n $ , $ A_{1},A_{2},\cdots,A_{n}\in F\left(S\right) $ , then $ \bigcup_{i=1}^{n}A_{i}\in F\left(S\right) $ .
3. If $ A_{i}\in F\left(S\right) $ , $ i=1,2,\cdots $ , then $ \bigcup_{i=1}^{\infty}A_{i}\in F\left(S\right) $ .
• A set $ F\left(S\right) $ with these 3 properties is called a sigma-field ($ \sigma $-field). If only 1 and 2 are satisfied, we have a field.
• It follows from three properties that $ \varnothing,S\in F\left(S\right) $ .
– Suppose $ A\in F\left(S\right) $ , then $ \bar{A}\in F\left(S\right) $ , $ A\cup\bar{A}=S\in F\left(S\right) $ , and $ \bar{S}=\varnothing\in F\left(S\right) $ .
• What about intersection? Suppose $ A,B\in F\left(S\right) $ . Is $ A\cap B\in F\left(S\right) $ ?
– $ A\cap B=\overline{\overline{A\cap B}}=\overline{\overline{A}\cup\overline{B}}\in F\left(S\right) $ .
1.2.3 Axioms of probability
• The probability measure $ P\left(\cdot\right) $ corresponding to $ S $ and $ F\left(S\right) $ is the assignment of a real number $ P\left(A\right) $ to each $ A\in F\left(S\right) $ satisfying following properties. Axioms of probability:
1. $ P\left(A\right)\geq0 $ , $ \forall A\in F\left(S\right) $ .
2. $ P\left(S\right)=1 $ .
3. If $ A_{1} $ and $ A_{2} $ are disjoint events, then $ P\left(A_{1}\cup A_{2}\right)=P\left(A_{1}\right)+P\left(A_{2}\right) $ . If $ A_{1},A_{2}\in F\left(S\right) $ and $ A_{1}\cap A_{2}=\varnothing $ , then $ A_{1} $ and $ A_{2} $ are disjoint events.
4. If $ A_{1},A_{2},\cdots,A_{n},\cdots\in F\left(S\right) $ is a countable collection of disjointed events, then $ P\left(\bigcup_{i=1}^{\infty}A_{i}\right)=\sum_{i=1}^{\infty}P\left(A_{i}\right) $ .
• $ P\left(\cdot\right) $ is a set function. $ P\left(\cdot\right):F\left(S\right)\rightarrow\mathbf{R} $ .
• If you want to talk about the probability of a single output $ \omega_{0}\in S $ , you do so by considering the single event
