Using classical probability to guarantee properties of infinite quantum sequences

We consider the product of infinitely many copies of a spin-1/2 system. We construct projection operators on the corresponding nonseparable Hilbert space which measure whether the outcome of an infinite sequence of σ^x measurements has any specified property. In many cases, product states are eigenstates of the projections, and therefore the result of measuring the property is determined. Thus we obtain a nonprobabilistic quantum analogue to the law of large numbers, the randomness property, and all other familiar almost-sure theorems of classical probability.