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In a probability theory script, I read a definition of independence where I don't understand one detail (Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space)

"A family of events $(A_i)_{i \in I}$, $A_i \in \mathcal{F}$ is called independent if $\{A_i, \Omega\}$ [*] with $i\in I$ is independent." (def. 7.2.1c)

Just before there is definied: "Subsets $\mathcal{E}_i$, $i \in I$ of $\mathcal{F}$ are called independent if all finite combinations of them are independent." (which comes down to the usual formula $\mathbb{P}(A \cap B) = \mathbb{P}(A) \mathbb{P}(B)$)

Why do I have to introduce the $\Omega$ in [*] ?

(The source is this script: http://www.wias-berlin.de/people/koenig/www/WTSkript.pdf)

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    Yes, I wrote it too short, actually I meant this definition also Stefan Hansen gave.."The sets (Ai)i∈I is said to be independent if that for every 1≤n≤|I| and choice of indices i1,…,in∈I we have that P(Ai1∩⋯∩Ain)=∏k=1nP(Aik)(∗)."2012-07-23

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The usual definition of independence of sets is the following.

Definition: The sets $(A_i)_{i\in I}$ is said to be independent if that for every $1\leq n\leq |I|$ and choice of indices $i_1,\ldots,i_n \in I $ we have that $ P(A_{i_1}\cap\cdots\cap A_{i_n})=\prod_{k=1}^n P(A_{i_k}) \qquad (*). $

If one were to check the definition in $(*)$, this would involve computing products with $2,3,\ldots, |I|$ terms. Therefore it is sometimes more convenient to introduce an equivalent characterization.

The sets $(A_i)_{i\in I}$ are independent if and only if $ P(B_1\cap\cdots \cap B_{N})=\prod_{k=1}^N P(B_k), \qquad (\text{here } N=|I|), $ for every choice $B_1,\ldots,B_N$ where $B_i\in \{A_i,\Omega\}$.

Note that this is again equivalent to saying that the sets $(\{A_i,\Omega\})_{i\in I}$ are independent.

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    There is also another version that comes in handy at times, though it is slightly more work to check. In the $2^N$ equations you state, change "$B_i \in \{A_i, \Omega\}$" to "$B_i \in \{A_i, A_i^c\}$"2012-07-23