Sentence examples for measurable partition of from inspiring English sources

If (alpha= { A_{1},A_{2},ldots,A_{n} }) is a measurable partition of (Omega_{2}), then (changing it on a set of measure zero if necessary) it is also a measurable partition of (X_{2}).

Let α be a measurable partition of Ω.

Let ((Omega,S,mu,T)) be a dynamical system and α be a measurable partition of Ω.

The inverse image (psi^{ - 1}alpha= { psi^{ - 1}(A_{i}); A_{i} inalpha }) is a measurable partition of (X_{1}) and hence of (Omega_{1}).

Each finite sequence ({ A_{1},A_{2},ldots, A_{n} }) of pairwise disjoint measurable subsets of Ω such that (bigcup_{i = 1}^{ n} A_{i} = Omega) is called a (measurable) partition of Ω. ([19]).

If we put (beta= { A_{i} cap Q, A_{i} cap Q^{C}, i = 1,2,ldots,n }), where Q is the set of all rational numbers in the real line (R^{1}), and (Q^{C} denotes the complement of Q, then β is a measurable partition of Ω.

The relation ≈ is an equivalence relation in the family of all measurable partitions of Ω.

Let ((Omega,S,mu,T)) be a dynamical system and α and β be measurable partitions of Ω such that (alpha precbeta).

3 and 4. In Sect. 3 we introduced the notions of logical entropy and logical conditional entropy of finite measurable partitions of a probability space and we examined basic properties of the proposed measures.

Consider the probability space ((Omega, S, mu)), and the measurable partition α of Ω from the previous example.

4, using the concept of logical entropy of measurable partitions, the notion of logical entropy of a dynamical system is introduced.