{
  "assumptions": [
    "α and β are measurable spaces; β is nonempty and standard Borel. μ is any finite nonnegative measure on α×β, including the zero measure. No standard-Borel or nonempty assumption is imposed on α.",
    "Let μ₁ be the first marginal and κ(a,db) a selected measurable regular conditional Markov kernel for the second coordinate given the first. You may use the existing finite-measure disintegration identity \\[\\mu(da,db)=\\mu_1(da)\\kappa(a,db).\\] Its conditional characterization is only μ₁-almost everywhere; the selected version has mass one at every input.",
    "The actual update retains a and resamples b: \\[K((a,b),d(a',b'))=\\delta_a(da')\\kappa(a,db').\\] Invariant means the measure equality μK=μ, not convergence from every starting measure."
  ],
  "binding_sha256": "3d72820e315cae6cebacdbed3bc8475fb4479d4a4a6cc7c302f4d534b54fe173",
  "domain": "mcmc",
  "family": "conditional-resampling-invariance",
  "id": "sogb-021",
  "level": "L3",
  "library_domains": [
    "mcmc",
    "discrete-sampling"
  ],
  "question": "Prove μK=μ by identifying the measure after projecting to the retained coordinate and after applying the product kernel. Explain the role of measurability, the finite-measure disintegration, and the chosen conditional version on null fibers. Must μ be a probability measure? Give a finite-state example showing why this invariance result alone does not imply convergence from another initial probability law.",
  "skills": [
    "proof",
    "assumption-audit",
    "counterexample"
  ],
  "split": "public-dev",
  "title": "Why retaining one coordinate and conditionally resampling the other is invariant",
  "track": "natural-language"
}