0.1 Probability Spaces
A sample space \(\Omega\) is the set of outcomes of a random experiment.
A \(\sigma\)-algebra \(\mathcal{F}\) is a collection of subspaces of \(\Omega\) satisfying the following properties: 1. \(\Omega \in\mathcal{F}\) 2. \(A\in\mathcal{F} \implies A^c\in\mathcal{F}\) (closed under compliments) 3. \(A_{i}\in\mathcal{F}\) for countable \(i\) \(\implies \cup_{i}A_{i}\in\mathcal{F}\) (closed under countable unions)
A \(\sigma\)-algebra is the collection of all measurable subsets of \(\Omega\) hence why we say the combination of \((\Omega,\mathcal{F})\) is a measurable space.
A probability measure \(\mathbb{P}\) is a mapping \(\mathbb{P}:\mathcal{F}\mapsto [0,1]\) satisfying the following properties: 1. \(\mathbb{P}(A)\geq \mathbb{P}(\emptyset)=0\); 2. \(\mathbb{P}(\Omega)=1\); 3. For countable disjoint \(A_{i}\in\mathcal{F}\) then \(\mathbb{P}(\cup_{i}A_{i})=\sum_{i}\mathbb{P}(A_{i})\).
A probability measure is a mapping assigning a measure on the interval \([0,1]\) for each measurable set. For measure \(\mathbb{P}\) on probability space \((\Omega, \mathcal{F}, \mathbb{P})\) we have the following results:
- Monotonicity: \(A\subset B\implies \mathbb{P}(A)\leq \mathbb{P}(B)\);
- Subadditivity: \(A\subset \cup_{m=1}^\infty A_{m}\implies P(A)\leq \sum_{m=1}^\infty \mathbb{P}(A_{m})\);
- Continuity from below: \(A_{i}\uparrow A\implies \mathbb{P}(A_{i})\uparrow \mathbb{P}(A)\);
- Continuity from above: \(A_{i}\downarrow A\implies \mathbb{P}(A_{i})\downarrow \mathbb{P}(A)\).
0.2 Borel \(\sigma\)-Algebra
Given any set \(A \in \mathcal{F}\) we can construct a \(\sigma\)-algebra by taking the intersection of all \(\sigma\)-algebra’s containing the set \(A\) \[ \sigma(A)=\bigcap_{\{\mathcal{F}:A\in \mathcal{F}\}}\mathcal{F}, \] which itself must also be a \(\sigma\)-algebra.
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