by
  rw [action_add_one_eq]
  have h_fun ω := algFunction_map alg n (hist alg ω n)
  refine ⟨by fun_prop, ?_⟩
  have h_indep : (arrayMeasure ν).map (fun ω ↦ (ω.1 (n + 1), hist alg ω n)) =
      (ℙ).prod ((arrayMeasure ν).map (hist alg · n)) := by
    have h_indep' := indepFun_fst_add_one_hist alg ν n
    rw [indepFun_iff_map_prod_eq_prod_map_map (by fun_prop) (by fun_prop)] at h_indep'
    rw [h_indep']
    congr
    simp only [arrayMeasure]
    calc ((Measure.infinitePi fun x ↦ ℙ).prod (streamMeasure ν)).map (fun ω ↦ ω.1 (n + 1))
    _ = (Measure.infinitePi fun x ↦ ℙ).map (Function.eval (n + 1)) := by
      nth_rw 2 [← Measure.fst_prod (μ := Measure.infinitePi fun x ↦ ℙ)
        (ν := streamMeasure ν)]
      rw [Measure.fst, Measure.map_map (by fun_prop) (by fun_prop)]
      rfl
    _ = ℙ := by rw [Measure.infinitePi_map_eval]
  have : (fun x ↦ (hist alg x n, algFunction alg n (hist alg x n) (x.1 (n + 1)))) =
      (fun p ↦ (p.2, algFunction alg n (p.2) (p.1))) ∘ (fun x ↦ (x.1 (n + 1), hist alg x n)) := rfl
  rw [this, ← Measure.map_map (by fun_prop) (by fun_prop), h_indep]
  have : (ℙ : Measure I).prod ((arrayMeasure ν).map (hist alg · n)) =
      ((Kernel.const _ ℙ) ×ₖ Kernel.id) ∘ₘ ((arrayMeasure ν).map (hist alg · n)) := by
    have h := Measure.compProd_const (μ := (arrayMeasure ν).map (hist alg · n))
      (ν := (ℙ : Measure I))
    rw [Measure.compProd_eq_comp_prod] at h
    rw [← Measure.prod_swap, ← h, ← Measure.deterministic_comp_eq_map (by fun_prop),
      Measure.comp_assoc, ← Kernel.swap, Kernel.swap_prod]
  rw [this, ← Measure.deterministic_comp_eq_map (by fun_prop),
    ← Measure.deterministic_comp_eq_map (by fun_prop), Measure.compProd_eq_comp_prod,
    Measure.comp_assoc, Measure.comp_assoc, Measure.comp_assoc]
  congr 2
  ext ω : 1
  simp only [Kernel.deterministic_comp_eq_map, Kernel.comp_deterministic_eq_comap, Kernel.coe_comap,
    Function.comp_apply]
  rw [Kernel.map_apply _ (by fun_prop), Kernel.prod_apply, Kernel.const_apply, Kernel.id_apply,
    Kernel.prod_apply, Kernel.id_apply, ← h_fun]
  calc (((ℙ).prod (Measure.dirac (hist alg ω n)))).map (fun p ↦ (p.2, algFunction alg n p.2 p.1))
  _ = (((ℙ).prod (Measure.dirac (hist alg ω n))).map Prod.swap).map
      (fun p ↦ (p.1, algFunction alg n p.1 p.2)) := by
    rw [Measure.map_map (by fun_prop) (by fun_prop)]
    rfl
  _ = ((Measure.dirac (hist alg ω n)).prod ℙ).map (fun p ↦ (p.1, algFunction alg n p.1 p.2)) := by
    rw [Measure.prod_swap]
  _ = (Measure.dirac (hist alg ω n)).prod ((ℙ).map (algFunction alg n (hist alg ω n))) := by
    ext s hs
    rw [Measure.map_apply (by fun_prop) hs, Measure.prod_apply, lintegral_dirac, Measure.prod_apply,
      lintegral_dirac, Measure.map_apply (by fun_prop)]
    · congr
    · exact hs.preimage (by fun_prop)
    · exact hs
    · exact hs.preimage (by fun_prop)