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On an Invariant Borel Measure in Hilbert Space

G. Pantsulaia (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

An example of a nonzero σ-finite Borel measure μ with everywhere dense linear manifold μ of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space ℓ₂ such that μ and any shift μ ( a ) of μ by a vector a ∈ ℓ ₂ ∖ μ are neither equivalent nor orthogonal. This extends a result established in [7].

On Ordinary and Standard Lebesgue Measures on ℝ ∞

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

New concepts of Lebesgue measure on ℝ ∞ 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 ℝ ∞ for α = (1,1,...).

On the existence of the functional measure for 2D Yang-Mills theory

Robert Budzyński (1997)

Banach Center Publications

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.

On the isotropic constant of marginals

Grigoris Paouris (2012)

Studia Mathematica

We show that if μ₁, ..., μₘ are log-concave subgaussian or supergaussian probability measures in ℝ n i , i ≤ m, then for every F in the Grassmannian G N , n , where N = n₁ + ⋯ + nₘ and n< N, the isotropic constant of the marginal of the product of these measures, π F ( μ ₁ ⊗ ⋯ ⊗ μ ₘ ) , is bounded. This extends known results on bounds of the isotropic constant to a larger class of measures.

Operator-valued Feynman integral via conditional Feynman integrals on a function space

Dong Cho (2010)

Open Mathematics

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 Γ t x = e x p ∫ 0 t θ s , x s d η s ϕ x t x ∈ C r 0 , t , 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...

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