Dependent Probability Spaces

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“…Автору известен ряд работ по аналогичной тематике [2]- [9]. В настоящей работе предполагается, что вероятности…”

On independent sets in purely atomic probability spaces with geometric distribution

“…This result should be seen in view of the known fact (see Problem 50, Section 4.1 in [7]) that if A 1 , A 2 , ...., A n are independent non-trivial events of a sample space X then |X| ≥ 2 n . One can observe that in general Ω(m) is considerably smaller than log 2 m. It is worth mentioning that according to [5] the first paper to deal to this problem in uniform finite probability spaces is [9]. In their paper, Shiflett and Shultz [9] raise the question of the existence of spaces with no non-trivial independent pairs, called dependent probability spaces.…”

“…For uniform distributed probability spaces X, as a result of the work in [6] and [1], X is dependent if |X| a prime number and independent if |X| is composite. For denumerable sets X one can see the construction given in [5] or look at the Example 1.1 in [10]. For our spaces, the Example 1.1 does not apply and in fact, we will construct explicitly lots of independent sets.…”

“…P r (T 1 ∩ T 2 |B) = P r (T 1 |B)P r (T 2 |B). (5) At this point the construction we have in Theorem 2 can be repeated. As a result, regardless of what r is, we obtain an uncountable family of there events which are mutually independent in (N 0 , A, P r ).…”