An example of a nonzero σ-finite Borel measure μ with everywhere dense linear manifold of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space ℓ₂ such that μ and any shift of μ by a vector are neither equivalent nor orthogonal. This extends a result established in [7].
New concepts of Lebesgue measure on are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on for α = (1,1,...).
We prove the existence of the path-integral measure of two-dimensional Yang-Mills theory, as a probabilistic Radon measure on the "generalized orbit space" of gauge connections modulo gauge transformations, suitably completed following the approach of Ashtekar and Lewandowski.
We show that if μ₁, ..., μₘ are log-concave subgaussian or supergaussian probability measures in , i ≤ m, then for every F in the Grassmannian , where N = n₁ + ⋯ + nₘ and n< N, the isotropic constant of the marginal of the product of these measures, , is bounded. This extends known results on bounds of the isotropic constant to a larger class of measures.
Let C 0r [0; t] denote the analogue of the r-dimensional Wiener space, define X t: C r[0; t] → ℝ2r by X t (x) = (x(0); x(t)). In this paper, we introduce a simple formula for the conditional expectations with the conditioning function X t. Using this formula, we evaluate the conditional analytic Feynman integral for the functional , where η is a complex Borel measure on [0, t], and θ(s, ·) and φ are the Fourier-Stieltjes transforms of the complex Borel measures on ℝr. We then introduce an integral...