We only need the two following basic lemmas; only sketchs of proof are given, we refer to Jean Jacod’s book, Probability essentials, for more information.
Definition: is a family of independent events if for every finite subset ,
Lemma: Let be a probability measure and be a family of independent events. Then
Sketch of proof. is a decreasing sequence of events, hence
Lemma: If is a familiy of independent events, then so is.
Sketch of proof. Using inclusion-exclusion principle, prove by induction on that every subfamily of cardinality satisfies
For , we have
Theorem: Let . Then .
Proof. Let be a random variable and be a probability measure such that
for example, take and . Let be the event “ is divisible by “. Notice that
Therefore, for any primes ,
We deduce that is a family of independent events. Because if and only if is not divisible by any prime, we have