pi_withDensity_prod · Samplinglib

Plain-English statement

- A finite product measure tilted by a product density decomposes coordinatewise.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Lean statement

theorem pi_withDensity_prod {E : ι → Type*}
    {mE : ∀ i, MeasurableSpace (E i)} {μ : (i : ι) → Measure (E i)}
    [∀ i, SigmaFinite (μ i)] {f : (i : ι) → E i → ℝ≥0∞}
    (hf : ∀ i, Measurable (f i))
    [∀ i, SigmaFinite ((μ i).withDensity (f i))] :
    (Measure.pi μ).withDensity (fun x => ∏ i, f i (x i)) =
      Measure.pi (fun i => (μ i).withDensity (f i)) := by
  classical
  refine (Measure.pi_eq (μ := fun i => (μ i).withDensity (f i)) fun s hs => ?_).symm
  rw [withDensity_apply _ (MeasurableSet.univ_pi hs),
    ← lintegral_indicator (MeasurableSet.univ_pi hs)]
  have hIndic : ∀ x : (i : ι) → E i,
      (Set.univ.pi s).indicator (fun x => ∏ i, f i (x i)) x =
        ∏ i, (s i).indicator (f i) (x i) := by
    intro x
    by_cases hx : x ∈ Set.univ.pi s
    · rw [Set.indicator_of_mem hx]
      refine Finset.prod_congr rfl (fun i _ => ?_)
      rw [Set.indicator_of_mem (hx i (Set.mem_univ _))]
    · rw [Set.indicator_of_notMem hx]
      rw [Set.mem_univ_pi] at hx
      push Not at hx
      obtain ⟨i, hi⟩ := hx
      exact (Finset.prod_eq_zero (Finset.mem_univ i)
        (Set.indicator_of_notMem hi _)).symm
  simp_rw [hIndic]
  rw [lintegral_fintype_prod_eq_prod (fun i => (hf i).indicator (hs i))]
  refine Finset.prod_congr rfl (fun i _ => ?_)
  rw [lintegral_indicator (hs i), ← withDensity_apply _ (hs i)]

end PiWithDensity

/-- The total mass of a `withDensity` measure is the lintegral of the density. -/

Open AutoSamplingTheory/TechnicalLemmas/Measure/RadonNikodym.lean:92published source at 7bcd37294df1

Proof architecture

Chewi MEAS/DENS root: decompose finite product density tilts into coordinatewise withDensity measures

Lean proof walkthrough

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Why the statement has this shape

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Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.
Density statements retain normalization and absolute-continuity prerequisites.
A Gibbs expression is not automatically a probability law or an invariant law.