Measurable sensitivity

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...

Measurable Utility and the Measurable Choice Theorem

Measurable Transformations

Measurable Categories

We develop the theory of categories of measurable fields of Hilbert spaces and bounded fields of operators. We examine classes of functors and natural transformations with good measure theoretic properties, providing in the end a rigorous construction for the bicategory used in [4] and [3] as the basis for a representation theory of (Lie) 2-groups. Two important technical results are establishe...

Measurable Rectangles

We give an example of a measurable set E ⊆ R such that the set E ′ = {(x, y) : x + y ∈ E} is not in the σ-algebra generated by the rectangles with measurable sides. We also prove a stronger result that there exists an analytic (Σ11) set E such that E ′ is not in the σ-algebra generated by rectangles whose horizontal side is measurable and vertical side is arbitrary. The same results are true wh...