Does the law of large numbers imply anything about "almost all samples"?

Question

Page 71 of "Information Theory, A Tutorial Introduction" states the following:

More importantly, as the messages are allowed to get longer, the law of large numbers guarantees that almost all messages generated will contain $nP$ $1$s.

As I understand the law of large numbers (both of them) it makes a statement about averages. I'd expect the number of messages containing exactly $nP$ $1$s to actually be vanishingly small.

($n$ is the message length, $P$ the probability of a $1$)

Answer 1

The statement is indeed wrong if taken literally as stated. To make it even more ridiculous, what if $nP$ is not an integer? What is meant is, for large $n$, in nearly all samples the fraction of $1$s will be close to $P$.

Answer 2

I'd expect the number of messages containing exactly $Pn$ $1$s to actually be vanishingly small

You are right. It can even be zero (for example when $P$ is irrational)

The statement you quote is very imprecise at best, wrong at worst. What the author meant is "the probability of getting a message with a fraction of ones that differs from $P$ in more that some fixed positive number tends to zero".