On measurable relation

The Full Group of a Countable Measurable Equivalence Relation

We study the group of all "R-automorphisms" of a countable equivalence relation R on a standard Borel space, special Borel automorphisms whose graphs lie in R. We show that such a group always contains periodic maps of each order sufficient to generate R . A construction based on these periodic maps leads to totally nonperiodic R-automorphisms all of whose powers have disjoint graphs. The prese...

On Measurable Stochastic Processes

In recent years probability theory has been formulated mathematically as measure theory; in the case of stochastic processes depending upon a continuous parameter the measures considered are defined on certain subspaces of the space of all functions of a real variable.! This formulation of stochastic processes depending upon a continuous parameter gives rise to certain measurability problems, a...

On Measurable Sierpiński-zygmund Functions

It is proved that there exists a Sierpiński-Zygmund function, which is measurable with respect to a certain invariant extension of the Lebesgue measure on the real line R. Let E be a nonempty set and let f : E → R be a function. We say that f is absolutely nonmeasurable if f is nonmeasurable with respect to any nonzero σ-finite diffused (i.e., continuous) measure μ defined on a σ-algebra of sub...

On Continuity of Measurable Cocycles

It is proved that every measurable, non-vanishing cocycle defined on the product of (0,∞) and an arbitrary compact metric space is continuous. Some other sufficient conditions for continuity of a cocycle are also given. Consider functions F satisfying the translation equation F (s+ t, x) = F (t, F (s, x)) (T) and real or complex valued solutions of the equation G(s+ t, x) = G(s, x)G(t, F (s, x)...

Measurable Utility and the Measurable Choice Theorem