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Existence of the independent Gaussian-marked positional lift
Statement
Assume AC. Let be any probability law on the simple locally finite positional configuration space of Poisson Configuration Reference, and let be a probability on the momentum mark space . Use the marked configuration space and local restrictions of Marked periodic continuation and local restriction. Define translations by and . Then there is a unique probability law on whose positional projection is and whose marks, conditional on the entire positional configuration, are independent with common law . If is translation invariant, meaning for every , then for every . For the Gaussian of Interacting marked phase-space and reference measures, this is the independent Gaussian-marked positional lift. This is a mathematical marking construction, not a physical preparation or a convergence result.
Facts & Assumptions
Given: A probability law P on Ω and a probability mark law ν on R^d.
AC is required by the standard-Borel coordinate extension used below (The Axiom of Choice).
Poisson Configuration Reference: Ω has countably many unit-cell coordinates; each cell is a finite configuration represented by its count and lexicographically ordered point list, and its sigma-algebra is generated by Borel count maps.
Marked periodic continuation and local restriction: Ω^mark has the marked-count sigma-algebra, positional projection, and finite-region restrictions ξ_Δ used in the statement.
Gaussian mark normalization, moments, and exponential controls: in the marked model, the normalized Gaussian kernel ν_b is a probability on R^d.
Assuming the Axiom of Choice, Kolmogorov extension for arbitrary families of standard Borel coordinate spaces: the consistent finite-dimensional product laws ν^{⊗J}, for finite J in a countable index set, extend uniquely to a probability on the countable product.
For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique and Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: product with P exists, and section integrals of nonnegative measurable functions are measurable.
Proof
Given: The positional probability P and mark probability ν above.
Let I=Z^d×N index one potential mark slot for each cell and each within-cell rank. For finite J⊂I prescribe the mark law ν^{⊗J}. These laws are consistent under coordinate restriction and are probabilities because ν is a probability by the Given hypothesis, so [F4] gives a probability ν^I under [A1]. The product probability P⊗ν^I exists by [F5].
For each cell i, write its finite positional list as x_{i,1},...,x_{i,N_i}, in the measurable lexicographic code of [F1]. Define Ψ(q,p) by pairing x_{i,k} with p_{i,k} for every i and 1≤k≤N_i(q). Every bounded spatial region meets finitely many unit cells, and each list is finite, so Ψ(q,p) is locally finite and simple in its positions. By [F2], it belongs to Ω^mark. For bounded Borel spatial A and Borel momentum set C, its marked count is the finite sum Each summand is measurable in the cell code and mark coordinate by [F1, F4], so Ψ is measurable for the sigma-algebra generated by these counts from [F2]. Its pushforward of P⊗ν^I is the required law P^mark.
Given q, each occupied point uses a distinct coordinate of the product ν^I, so its finite mark families have product law ν^{⊗J}, independent of q. This proves the conditional independence property and fixes every finite marked-count cylinder probability. Those cylinders generate the marked configuration sigma-algebra by [F2], so any law with the same position marginal and conditional product kernel agrees with the pushforward in step 2.1 on a generating class and is identical. Thus the lift exists and is unique. By [F3], the Gaussian kernel ν_b of the model is an admissible probability choice for ν. For each y, the map T_y^mark is measurable because it sends the count N_{A×C} to N_{(A-y)×C}. If P is translation invariant, translating the constructed law by y leaves its positional marginal P and its conditional independent mark kernel ν unchanged; uniqueness therefore gives (T_y^mark)_#P^mark=P^mark.
Depends on
The Axiom of Choice
Interacting marked phase-space and reference measures
Marked periodic continuation and local restriction
Poisson Configuration Reference
Gaussian mark normalization, moments, and exponential controls
Assuming the Axiom of Choice, Kolmogorov extension for arbitrary families of standard Borel coordinate spaces
For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique
Tonelli's theorem for nonnegative measurable functions on a sigma-finite product