CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Canonical Markov chain on path space

Statement

Assume Choice. For every probability measure $μ$ and probability kernel $K$ on an arbitrary measurable space $(E, E)$ , the canonical path space $(E^{N_{0}}, E^{\otimes N_{0}})$ carries a unique probability $P_{μ}$ under which the coordinate maps form a Markov chain with initial law $μ$ and transition kernel $K$ .

Facts & Assumptions

Given: Choice, $(E, E)$ , $μ$ , and $K$ as in the statement.

[F1]

Ionescu--Tulcea constructs a unique countable-product law for specified history-dependent kernels, and its homogeneous last-coordinate specialization is a Markov chain. (Ionescu-Tulcea construction of a Markov chain)

Proof

1.1

In [F1], take $E_{n} = E$ for all $n$ , initial law $μ$ , and [F1] $K_{n} (x_{0}, \dots, x_{n}, A) = K (x_{n}, A) .$ The last display is a probability kernel because it is the composition of the measurable last-coordinate projection with each measurable evaluation of $K$ .

2.1

The theorem [F1] therefore gives a unique law on the product sigma-algebra and says that [F1, step 1.1] the coordinates are the required $(μ, K)$ -chain. Choice is exactly the assumption of [F1]. The construction includes Dirac initial laws, one-point spaces, and $n = 0$ ; an empty $E$ admits no probability $μ$ and so cannot meet the hypotheses.

F1step 1.1∎

Depends on

Used by

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Shalizi, Building Infinite Processes from Finite-Dimensional Distributions, Theorem 33 (standard reference, not scraped)