{
  "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": "L2",
  "library_domains": [
    "mcmc",
    "discrete-sampling"
  ],
  "question": "Complete the fixed Lean target below. Keep its imports, context and proposition unchanged. Use the repository-pinned Lean/Mathlib. This is open-library reconstruction; the target module is not imported directly.\n\n/- SamplingOptimisationGeometryBench public-dev reference.\n   Reuses the existing project proof; retain LICENSE and NOTICE. -/\nimport AutoSamplingTheory.TechnicalLemmas.Probability.ConditionalResampling\nimport Mathlib.Probability.Kernel.Invariance\n\nopen MeasureTheory ProbabilityTheory\nnamespace AutoSamplingTheory.TechnicalLemmas.Probability.HeatBath\nnoncomputable def heatBathSnd {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]\n    [StandardBorelSpace β] [Nonempty β] (μ : Measure (α × β)) [IsFiniteMeasure μ] :\n    Kernel (α × β) (α × β) :=\n  (Kernel.id ×ₖ condDistrib Prod.snd Prod.fst μ) ∘ₖ\n    Kernel.deterministic Prod.fst measurable_fst\nend AutoSamplingTheory.TechnicalLemmas.Probability.HeatBath\nopen AutoSamplingTheory.TechnicalLemmas.Probability.HeatBath\nopen AutoSamplingTheory.TechnicalLemmas.Probability\nvariable {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]\n  [StandardBorelSpace β] [Nonempty β]\n\nexample (μ : Measure (α × β)) [IsFiniteMeasure μ] :\n    (heatBathSnd μ).Invariant μ := by\n  -- Replace this comment with your proof.\n",
  "skills": [
    "proof",
    "assumption-audit",
    "counterexample"
  ],
  "split": "public-dev",
  "title": "Why retaining one coordinate and conditionally resampling the other is invariant",
  "track": "lean"
}