Orthogonal series and probability
Dan Eustice · Proceedings of the American Mathematical Society · 1967
2. Definitions. Let (X, 8, IA) be a probability space. [See [5] for probability theory concepts.] An orthonormal system {1. } on (X, 8, y) is called an H-system if the following conditions are satisfied: (1) Each 4. assumes at most two nonzero values with positive probability, (2) the a-field generated by {}f>, written a(41, 2, O *N) consists of exactly N atoms, (3) E(4n+1I41, * , ) =9 0 n_ 1, that is, the conditional expectation of On+1 with respect to u(4l, . . . , 4n) is zero. In [1] it is shown that this is equivalent to {q5, } having the property, that for every f in L2(X),