Conditional zero-one laws

We say that a class of events fulfills a conditional zero-one law if it is a subset of the completion of the conditioning $\sigma$-algebra. In this case the conditional probability of an event of the class is an indicator function. Therefore the conditional probability takes almost surely only the values zero and one; in the unconditional case the indicator functions are almost surely constant.
We consider two special zero-one laws. If a sequence of random variables is conditionally independent, then its tail $\sigma$-algebra fulfills a conditional zero-one law; this generalizes Kolmogorov's zero-one law. If the sequence is even conditionally identically distributed, then its permutable $\sigma$-algebra, which contains the tail $\sigma$-algebra, fulfills a conditional zero-one law; this generalizes the zero-one law of Hewitt and Savage.