Exact source context

import Mathlib.Probability.Kernel.CondDistrib

/-!
# Conditional resampling of a joint law

This file gives the product-state specialization of Mathlib's disintegration
identity that is needed before Gibbs/heat-bath updates are packaged as Markov
kernels.  It does not define a Gibbs sampler or prove a mixing rate.
-/

namespace AutoSamplingTheory
namespace TechnicalLemmas
namespace Probability
namespace ConditionalResampling

open MeasureTheory
open scoped ProbabilityTheory

/-- A joint finite law is recovered by combining its first marginal with the
conditional distribution of the second coordinate given the first.

This is the source-neutral law identity behind one-block Gibbs/heat-bath
resampling.  It says nothing about scan order, irreducibility, convergence, or
mixing; those are downstream kernel obligations.

The nonempty Standard Borel hypothesis concerns the resampled coordinate `β`,
not `α`. It is the regular-conditional-distribution existence contract used by
Mathlib; the selected version agrees with the conditional law only almost
everywhere for the first marginal. No conditional-support assertion on null
fibers is made. Finite measures, including the zero measure, are allowed: this
is not restricted to probability laws. -/
theorem fst_compProd_condDistrib_snd_eq_self
    {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
    [StandardBorelSpace β] [Nonempty β]
    {μ : Measure (α × β)} [IsFiniteMeasure μ] :
    (μ.map Prod.fst) ⊗ₘ
        ProbabilityTheory.condDistrib Prod.snd Prod.fst μ = μ := by
  have h := ProbabilityTheory.compProd_map_condDistrib
    (μ := μ) (X := Prod.fst) (Y := Prod.snd) (mβ := inferInstance)
    measurable_snd.aemeasurable
  simpa using h

end ConditionalResampling
end Probability
end TechnicalLemmas
end AutoSamplingTheory