Talk:Kolmogorov's zero–one law

message: First, I think you need to say more about the definition of a tail event, because consider:
Let X_i(\omega)\ :\ i \in \mathbb{N} be i.i.d. Bernoulli random variables on \Omega satisfying \operatorname{P}(X_i = 1) = 1/2. Define I_X \subset \mathbb{N} as the set of all i \in \mathbb{N} such that X_i = 1. Using the axiom of choice, choose a family \mathcal{A} of subsets of \mathbb{N} such that for each B \subset \mathbb{N} there exists unique A \in \mathcal{A} such that the symmetric difference of A and B is finite. Now let \mathcal{A}' be a subset of \mathcal{A} of cardinality 2^{\aleph_0} whose complement is also of cardinality 2^{\aleph_0}. Let Z be the set \{\omega \in \Omega : \exists A \in \mathcal{A}' \textrm{s.t.}\,I_X \bigtriangleup A\,\textrm{is}\,\textrm{finite}\}. Let \mathcal{F} be the σ-algebra on \Omega generated by the X_i and let \mathcal{G} be the σ-algebra generated by \mathcal{F} \cup \{Z\}. An arbitrary element of \mathcal{G} can be expressed in the form (A\cap Z)\cup (B\cap Z^c) where A,B \in \mathcal{F}. Define the probability of such an event to be (\mathop{P}(A) + \mathop{P}(B)) / 2. Then the event Z is independent of all finite subsets of the X_i and is uniquely determined by the 'tail' (X_n, X_{n+1}, X_{n+2}, \ldots) for any n, but has probability 1/2.
The reason why this doesn't contradict the theorem is because the theorem should only be stated for events belonging to \mathcal{F}.
Another issue is that you <I>define</I> a tail event as being one that is independent of all finite subsets of the X_i, but this itself follows from the hypothesis that the X_i are independent. In fact, under the article's definition of 'tail event', surely we don't even need to assume that the X_i are independent for the result to be true (provided it's only stated for events in \mathcal{F})?  —Preceding unsigned comment added by 92.15.133.253 (talk) 10:38, 6 August 2010 (UTC)
thread_title: A couple of issues I have with this article

message: This article lacks a proof. 240F:7C:FC1A:1:6505:ECD6:F741:7B7E (talk) 01:06, 20 December 2014 (UTC)
thread_title:  More information needed 

message: An invertible measure-preserving transformation on a standard probability space that obeys the 0-1 law is called a Kolmogorov automorphism.
What "obeys" the 0-1 law? The space? The transformation? Neither makes sense. Indeed, it's hard to see how a thing can "obey" the 0-1 law since the 0-1 law is not formulated as a property. 76.118.180.76 (talk) 02:00, 15 December 2015 (UTC)
thread_title:  Unclear sentence 

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thread_title:  External links modified