Two events \(A,B\in \mathcal{F}\) are independent if \(P(A\cap B)=P(A)P(B)\).
Events \(A_1,\ldots A_n\) are independent if \[ P(A_{i_1}\cap\cdots \cap A_{i_m})=P(A_{i_1})\cdots P(A_{i_m}) \] for all \(2\le m\le n\) and \(1\le i_1<\cdots<i_m\le n\).
Events \(A_1,\ldots,A_n\) are independent if and only if \[ P(B_1\cap\cdots B_n)=P(B_1)\cdots P(B_n) \] for all \(B_1,\ldots,B_n\) where for each \(i=1,\ldots,n\), either \(B_i=A_i\) or \(B_i=\Omega\).
Events \(A_t, t\in T\) are independent if for every finite collection of distinct indicies \(\{t_1,\ldots,t_n\}\subset T\), \[ P(A_{t_1}\cap\cdots\cap A_{t_n})=P(A_{t_1})\cdots P(A_{t_n}). \] Equivalently, events \(A_t,t\in T\) are independent if and only if the events in every finite subcollection are independent.
Classes of events \(\mathcal{A}_1,\ldots,\mathcal{A}_n\) are independent if for every choice of events \(A_i\in \mathcal{A}_i\), \(i=1,\ldots,n\), the events \(A_1,\ldots,A_n\) are independent.
Classes of events \(\mathcal{A}_1,\ldots,\mathcal{A}_n\) are independent if and only if \[ P(B_1\cap\cdots\cap B_n)=P(B_1)\cdots P(B_n) \tag{*} \] for all \(B_i\in \mathcal{B}_i, i=1,\ldots,n\), where \(\mathcal{B}_i=\mathcal{A}_i\cup\{\Omega\}\).
If \(\mathcal{A}_1,\ldots,\mathcal{A}_n\) are independent \(\pi\)-systems, then \(\sigma(\mathcal{A}_1),\ldots,\sigma(\mathcal{A}_n)\) are independent \(\sigma\)-fields.
\(\pi-\lambda\) theorem으로 증명한다.
\(\mathcal{B}_i=\mathcal{A}_i\cup \Omega\)라 하자. \(\mathcal{L}:=\{A\in \mathcal{F}:P(A)\cap P(B_2)\cdots\cap P(B_n)=P(A)P(B_2)\cdots P(B_n)\}\)일 때 이는 \(\lambda\)-system이고, \(\mathcal{B}_1\in \mathcal{L}\)이다. \(\mathcal{B}_1\)은 \(\pi\)-system이기 때문에 때문에 \(\sigma(\mathcal{A}_1)=\sigma(\mathcal{B}_1)\subset \mathcal{L}\)이다. 또한 \(\mathcal{L}\)안에서의 \(A,B_2,\ldots,B_n\)은 모두 arbitrary set in each class이다. 때문에 \(\sigma(\mathcal{A}_1)=\subset \mathcal{L}\)이므로\(\sigma(\mathcal{A}_1),\ldots,\sigma(\mathcal{A}_n)\)안의 모든 초이스들에 대해서 independent하다.
Classes of events \(\mathcal{A}_t, t\in T\) are independent if for every finite collection of distinct indicies, \(t_1,\ldots,t_n\in T\) and events \(A_{t_i}\in \mathcal{A}_{t_i}, i=1,\ldots,n\), the events \(A_{t_1},\ldots A_{t_n}\) are independent.
If \(\mathcal{A}_t,t\in T\) are independent \(\pi\)-systems, then \(\sigma(\mathcal{A}_t),t\in T\) are independent \(\sigma\)-fields.