Probability Measure

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Probability Measure

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Definition

MeasureTheory

A finitely additive probability measure on $(\Omega,\mathcal{B}_{0})$ , (where $\mathcal{B}_{0}$ is an algebra) is a map $\mathbb{P}:\mathcal{B}_{0}\to[0,1]$ such that $\mathbb{P}(\emptyset)=0,\quad\mathbb{P}(\Omega)=1$ and $A,B\in\mathcal{B}_{0},A\cap B=\emptyset\implies \mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)$

Let $(\Omega,\mathcal{F})$ be a measurable space, a probability measure $\mathbb{P}$ is a measure such that $\mathbb{P}(\Omega)=1$

For $A\in\mathcal{F}$ : $\mathbb{P}(A^{c})=1-\mathbb{P}(a)$

For $A,B\in\mathcal{F}$ , if $A\subseteq B$ then $\mathbb{P}(A)\le \mathbb{P}(B)$

For 2 events, let $A_1,A_2\in\mathcal{F}$ . $P(A_1\cup A_2)=P(A_1)+P(A_2)-P(A_1\cap A_2)$ For $n$ events, let $A_1,...,A_n\in\mathcal{F}$ : $P(\bigcup^n_{i=1}A_i)=\sum^{n}_{k=1}(-1)^{k-1}\sum_{1\le i_1<\ldots< i_k\le n}P(A_{i_1}\cap \ldots\cap A_{i_k})$

For $(A_{n})_{n\ge 1}\subseteq \mathcal{F}$ we have that $\mathbb{P}\left( \bigcup_{n\ge 1} A_{n} \right)\le \sum_{n\ge 1}\mathbb{P}(A_{n})$

Probability Measure

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