Measurable transformations

Measurable Transformations

Spaces of Measurable Transformations

By a space we shall mean a measurable space, i.e. an abstract set together with a <r-ring of subsets, called measurable sets, whose union is the whole space. The structure of a space will be the <r-ring of its measurable subsets. A measurable transformation from one space to another is a mapping such that the inverse image of every measurable set is measurable. Let X and F be spaces, F a set of...

Measurable Utility and the Measurable Choice Theorem

The Approximation of One-one Measurable Transformations by Measure Preserving Homeomorphisms

This paper contains two results related to the material in [2]. Suppose/is a one-one transformation of the open unit interval /" (where n^2) onto /". 1. If/ is absolutely measureable and £>0, then there is an absolutely measurable homeomorphism ç»t of /" onto /" such that m({x:f(x)^(pc(x) or f~'(x)^ 9'J1W})<£. where m denotes /»-dimensional Lebesgue measure. 2. Suppose p is either (1) a nonatom...

Measurable Sensitivity

We introduce the notions of measurable and strong measurable sensitivity, which are measure-theoretic versions of the conditions of sensitive dependence on initial conditions and strong sensitive dependence on initial conditions, respectively. Strong measurable sensitivity is a consequence of light mixing, implies that a transformation has only finitely many eigenvalues, and does not exist in t...