Kolmogorov's zero–one law

The Kolmogorowsches zero-one law, named after the Russian mathematician Andrei Nikolaevich Kolmogorov, is a set from the field of probability theory. Like all zero-one laws he describes a class of events whose probabilities are or always.

Formulation

Let be a probability space, and, a sequence of independent σ - algebras. Identifies the associated terminal σ - algebra, so is this - trivial, ie for everyone.

Evidence

It suffices to show that is independent of itself, because then applies, ie. According to the principle of good quantities to be the set of all independent events. Then is a Dynkin system. As is by definition measurable, includes the - stable systems, so even an algebra. From the additivity follows that. Since, it follows that.

Application

Be independent random variables, and the corresponding terminal to algebra. It is easy, that. The sequence converges or diverges almost certainly so. Inscribed in the first case the limit, it can be further show that a - measurable random variable. Since it is trivial, must be constant necessary.