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On a probability space, from Billingsley's Probability and Measure,

a family of classes of events is said to be independent, if we arbitrarily choose an event from each class in the family, and these events are then independent.

According to the above definition, I deduce the following definition:

An event $A$ is said to be independent of events $B_1, B_2,\dots , B_m$ if $A$ and $B_i$ are independent, $\forall i =1, \dots, m$.

However, I saw from someone's private note that

An event $A$ is said to be independent of events $B_1, B_2,\dots , B_m$ if for evew subset $S$ of $\{1, 2,\dots ,m\}$, $$ P[A |\cap_{i \in S} B_i] = P[A]. $$

  1. Are the two different definitions equivalent?

    In other words, is the following true:

    $A$ and $B_i$ are independent, $\forall i =1, \dots, m$, if and only if for evew subset $S$ of $\{1, 2,\dots ,m\}$, $ P[A |\cap_{i \in S} B_i] = P[A]. $

  2. I wonder how to define independence between an event and a set of events correctly?
  3. Same questions when there are infinitely many events $B_i, i \in I$?
  4. Added: What is the definition of independence of an event and a set of events in Lovasz local lemma?

Thanks!

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1 Answer

up vote 3 down vote accepted

The two are not equivalent (I suspect the first definition may have been meant for a more specific context.)

Easy counterexample are variables $B_1,B_2$ that independently take values $-1$ and $1$ with chance half and $A=B_1B_2$. Then $A$ is independent of $B_1$ and $B_2$ separately but is fully determined by the two of them.

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+1, though your counter example is of random variables rather than events. –  Henry Oct 11 '12 at 18:05
    
Thanks! +1. What is the definition of independence of an event and a set of events in Lovasz local lemma? –  Tim Oct 11 '12 at 18:25
    
@Tim, the second definition is standard and looking around seems to confirm that it is indeed the definition used in the Lovasz local lemma. –  anonymous Oct 11 '12 at 19:28
    
Thanks! WHat sources have you found? –  Tim Oct 11 '12 at 19:38
    
cse.buffalo.edu/~hungngo/classes/2011/Spring-694/lectures/… was fairly explicit w.r.t. to independence. –  anonymous Oct 11 '12 at 20:36
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