(Please read about tail events first)
Theorem.(explanations below) Suppose a probability space (Ω,B,P) is generated by a sequence of random variables X1,X2,... (that is, any event E∈B can be approximated by events {Xi1∈I1} ∩ ... ∩ {Xik∈Ik}; knowing the values of the Xi's is enough to know all there is to know about randomness in our sample space). Then for any tail event (which see) E, either P(E)=1 or P(E)=0 (either E occurs almost always or E does not occur, almost always).
Colloquially we might say that an event which is independent of X1,X2,... in a measure space which is specified by these random variables must occur with probability 0 or 1, since it is independent of everything in the measure space; the proof is just a technicality which says this.
- Consider an infinite random stream of fair coin tosses: an infinite sequence of 0's and 1's, chosen IID with equal probabilities at each step. Call the result of the i'th coin toss Xi (these variables generate the sample space). Does the Cesaro limit
limn->∞ (X1+...+Xn)/n
of the random variables exist?
This event (of the limit existing) is independent of the values of any finite number of X's: existence of a limit or a Césaro limit does not depend on any finite prefix of the sequence. So Kolmogorov's 0-1 law says either the limit exists with probability 1 or there is no limit with probability 1. In fact, the strong law of large numbers states that the limit exists, with probability 1.
- Consider the random graph we get from the square grid Z2 by keeping each edge with probability p (0<p<1 some predetermined constant) IID, and deleting it otherwise (this is percolation on the square grid, and has been extensively studied). Does there remain an infinite connected component somewhere in the graph?
Any event that finitely many edges are removed cannot affect the existence of an infinite connected component -- it can only disconnect a finite number of vertices! So the event that there exists an infinite connected component is a tail event of the sequence of indicator variables of the edges remaining in the graph; by Kolmogorov's 0-1 law, for any p the probability that there exists an infinite connected component is either 0 or 1.
This result forms the cornerstone of the study of percolation.