On some properties of classes of events for which the conditions for the uniform convergence of the relative frequencies to probabilities fail to hold

We show that, if a system of events $S$ does not satisfy the conditions for the uniform convergence of the relative frequencies to probabilities, that is, if the limit entropy per symbol is greater than zero, then there is necessarily an event $T$ having the following two properties: if $x^l$ is an independent random sample and $x^l(T)$ is the part of $x^l$ belonging to $T$, then the system of events induces all possible subsamples on $x^l(T)$ with probability $1$, and the probability measure of $T$ is precisely equal to the limit entropy per symbol.