Schrödinger’s cat

Saturated Structures From Probability Theory

“…We write simply pr Γ if ∆ = κ. The following proposition describes the Borel measurable subsets of [0,1] κ .…”

“…[11,Theorem 3.9]). For all atomless probability spaces Ω, there is a countable set of distinct infinite cardinals S = {κ i | i ∈ I} such that the measure algebra of Ω is isomorphic to a convex combination of the homogeneous probability algebras [0, 1] κi . The set S is uniquely determined by Ω and is called the Maharam spectrum of Ω.…”

“…First, we introduce the theory APr, which was first studied by Ben Yaacov in [1] under the setting of compact abstract theories. Later, his work was carried into continuous logic in the background sections of Berenstein and Henson in [7].…”

“…is the probability algebra of ([0, 1] λ , µ λ ), every cardinal in the Maharam spectrum of B ′ is > λ, the real number α ∈ (0, 1], and µ ′ (1 B ′ ) = 1 − α. Note that A λ as a metric space with the metric d λ (a, b) := µ λ (a△b) has a dense subset of cardinality λ.…”

“…Also, we define a ([0, 1]-valued) random variable structure L 1 (Ω, F , µ), [0, 1] based on (Ω, F , µ); see more in Section 2.2. Ben Yaacov [1] studied probability algebras in the setting of compact abstract theories. His work on probability algebras was later carried into continuous logic by Berenstein and Henson as part of the background of their paper [7].…”