Paul Garrett [email protected] http://www.math.umn.edu/ g̃arrett/ [This document is http://www.math.umn.edu/ ̃garrett/m/fun/choice functions.pdf] This note is essentially a rewrite of Appendix V of J. Dixmier’s von Neumann Algebras, North Holland, 1981. The point is to obtain the practical special case: [0.0.1] Corollary: Let X be a locally compact Hausdorff second-countable [1] topological sp...

We make the first steps towards showing a general “randomness for free” theorem for stochastic automata. The goal of such theorems is to replace randomized schedulers by averages of pure schedulers. Here, we explore the case of measurable multifunctions and their measurable selections. This involves constructing probability measures on the measurable space of measurable selections of a given me...

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 (Σ1) 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 whe...

‎Let $M(X‎, ‎mathcal{A}‎, ‎mu)$ be the ring of real-valued measurable functions‎ ‎on a measure space $(X‎, ‎mathcal{A}‎, ‎mu)$‎. ‎In this paper‎, ‎we characterize the maximal ideals in the rings of real measurable functions‎ ‎and as a consequence‎, ‎we determine when $M(X‎, ‎mathcal{A}‎, ‎mu)$ is a hereditary ring.

We formulate variants of the cardinals f, p and t in terms of families of measurable functions, in order to examine the effect upon these cardinals of adding one random real.