The meaning of probability when large number law fails When the large number law fails, the frequency of an event does not converge to the probability, so what is the meaning of probability in these cases? 
PS: Please adds references to the answers (if it possible).
 A: As pointed out in the comments and as discussed here, it is extremely difficult to think of practical examples where the LLN fails to hold for the probability of an event. I think this illustrates the strength of Kolmogorov's result.
It is also important to point out that it pacified the debate between Bayesian and frequentists. It means that you do not need to worry about the question "what is a probability?", but only about "can I measure it?".
Still, there is one obvious case when it is impossible to get IID sampling of an event: when it happens only once. In that case the LLN does not apply and we are back to good old debate on what is a probability. But I think this is also outside the scope of this forum so I'll just cite some references from the top of my head.
Henri Poincaré in Science et Méthodes (1914) emphasized the deteministic nature of randomness. Chance is an effect of the imprecision of our measure instruments.
Paul Lévy in Calcul des probabilités (1925) advocates that probabilities of single events are meaningful only if they are close to 0 and 1.
Bruno de Finetti proposed to define probability as a coherent weighing scheme of prior beliefs.
More recently, Peter Walley in Statistical reasoning with imprecise probabilities tried to extend de Finetti's definition to make probability a representation of human knowledge.
This list is obviously very biased and incomplete to say the least.
A: I think you are confusing events with random variables.  The strong law of large numbers says that if $S_n= (X_1+X_2+X_3+...+X_n)$ then $S_n/n$ converges to a constant almost surely.  This is convergence of a sequence of random variables to a random variable that is degenerate (i.e. equal to a constant).  If the law of large numbers fails the convergence could be to a non-degenerate distribution or it may just not converge at all. But this does not connect to relative frequencies of measurable events in a probability space.  Suppose $A$ is a measureable event in a probability space.  Define the independent binary variables $X_i$ for $i=1,2,3,..,n$ such that $X_i=1$ if $A$ occurs and $X_i=0$ otherwise.  Then $S_n$ is Binomial in this case and $S_n/n$ does converge to $p=P(A)=E(X_i)$.  So there is no contradiction about measurable events having their probaility expressible as a limiting relative frequency.
A: The Law of Large numbers as it is commonly defined doesn't deal with events converging to a probability but with the results of events converging to an average value.
Let's take an example of a die with six sides.
According to the law of large numbers the average is supposed to converge towards $\frac{1+2+3+4+5+6}{6}=3.5$. A probability is a value between 0 and 1. The 3.5 in this example is no probability. The probability that a single dice hits 3.5 is even zero. 
