diff --git a/LeanBandits/Bandit.lean b/LeanBandits/Bandit.lean
index f5e6c131..07a3f5d4 100644
--- a/LeanBandits/Bandit.lean
+++ b/LeanBandits/Bandit.lean
@@ -15,11 +15,16 @@ open MeasureTheory ProbabilityTheory Filter Real Finset
 
 open scoped ENNReal NNReal
 
-/-- Measurable equivalence between `Iic 0 → α` and `α`. -/
-def MeasurableEquiv.piIicZero (α : Type*) [MeasurableSpace α] :
-    (Iic 0 → α) ≃ᵐ α :=
-  have : Unique (Iic 0) := by simp only [mem_Iic, nonpos_iff_eq_zero]; exact Unique.subtypeEq 0
-  MeasurableEquiv.funUnique _ _
+instance : Unique (Iic 0) := by simp only [mem_Iic, nonpos_iff_eq_zero]; exact Unique.subtypeEq 0
+
+lemma coe_default_Iic_zero : ((default : Iic 0) : ℕ) = 0 := by
+  calc _ = ((⟨0, by simp⟩ : Iic 0) : ℕ) := by congr; exact (Unique.eq_default _).symm
+    _ = _ := by simp
+
+/-- Measurable equivalence between `Iic 0 → X i` and `X 0`. -/
+def MeasurableEquiv.piIicZero (X : ℕ → Type*) [∀ n, MeasurableSpace (X n)] :
+    ((i : Iic 0) → X i) ≃ᵐ X 0 :=
+  (MeasurableEquiv.piUnique _).trans (coe_default_Iic_zero.symm ▸ MeasurableEquiv.refl _)
 
 namespace Bandits
 
@@ -39,6 +44,14 @@ structure Algorithm (α R : Type*) [MeasurableSpace α] [MeasurableSpace R] wher
 instance (alg : Algorithm α R) (n : ℕ) : IsMarkovKernel (alg.policy n) := alg.h_policy n
 instance (alg : Algorithm α R) : IsProbabilityMeasure alg.p0 := alg.hp0
 
+/-- A deterministic algorithm. -/
+noncomputable
+def detAlgorithm (nextArm : (n : ℕ) → (Iic n → α × R) → α) (h_next : ∀ n, Measurable (nextArm n))
+    (arm0 : α) :
+    Algorithm α R where
+  policy n := Kernel.deterministic (nextArm n) (h_next n)
+  p0 := Measure.dirac arm0
+
 namespace Bandit
 
 /-- Kernel describing the distribution of the next arm-reward pair given the history up to `n`. -/
@@ -71,7 +84,7 @@ deriving IsMarkovKernel
 /-- Measure on the sequence of arms pulled and rewards observed generated by the bandit. -/
 noncomputable
 def trajMeasure (alg : Algorithm α R) (ν : Kernel α R) [IsMarkovKernel ν] : Measure (ℕ → α × R) :=
-  (traj alg ν 0) ∘ₘ ((alg.p0 ⊗ₘ ν).map (MeasurableEquiv.piIicZero _).symm)
+  (traj alg ν 0) ∘ₘ ((alg.p0 ⊗ₘ ν).map (MeasurableEquiv.piIicZero (fun _ ↦ α × R)).symm)
 
 /-- Measure of an infinite stream of rewards from each arm. -/
 noncomputable
@@ -82,7 +95,8 @@ deriving IsProbabilityMeasure
 instance (alg : Algorithm α R) (ν : Kernel α R) [IsMarkovKernel ν] :
     IsProbabilityMeasure (trajMeasure alg ν) := by
   rw [trajMeasure]
-  have : IsProbabilityMeasure ((alg.p0 ⊗ₘ ν).map (MeasurableEquiv.piIicZero _).symm) :=
+  have : IsProbabilityMeasure
+      ((alg.p0 ⊗ₘ ν).map (MeasurableEquiv.piIicZero (fun _ ↦ α × R)).symm) :=
     isProbabilityMeasure_map <| by fun_prop
   infer_instance
 
@@ -130,22 +144,107 @@ lemma measurable_reward_prod : Measurable (fun p : ℕ × (ℕ → α × R) ↦
 @[fun_prop]
 lemma measurable_hist (n : ℕ) : Measurable (hist n (α := α) (R := R)) := by unfold hist; fun_prop
 
+lemma hist_eq_frestrictLe :
+    hist = Preorder.frestrictLe («π» := fun _ ↦ α × R) := by
+  ext n h i : 3
+  simp [hist, Preorder.frestrictLe]
+
 /-- Filtration of the bandit process. -/
 protected def filtration (α R : Type*) [MeasurableSpace α] [MeasurableSpace R] :
     Filtration ℕ (inferInstance : MeasurableSpace (ℕ → α × R)) :=
   MeasureTheory.Filtration.piLE (X := fun _ ↦ α × R)
 
+section Traj
+
+open Kernel Preorder
+
+variable {X : ℕ → Type*} [∀ n, MeasurableSpace (X n)]
+    {κ : (n : ℕ) → Kernel ((i : { x // x ∈ Iic n }) → X i) (X (n + 1))} [∀ n, IsMarkovKernel (κ n)]
+
+lemma Measure.compProd_map {X Y Z : Type*} {mX : MeasurableSpace X} {mY : MeasurableSpace Y}
+    {mZ : MeasurableSpace Z} {μ : Measure X} {κ : Kernel X Y} [SFinite μ] [IsSFiniteKernel κ]
+    {f : Y → Z} (hf : Measurable f) :
+    μ ⊗ₘ (κ.map f) = (μ ⊗ₘ κ).map (Prod.map id f) := by
+  calc μ ⊗ₘ (κ.map f)
+  _ = (Kernel.id ∥ₖ Kernel.deterministic f hf) ∘ₘ (Kernel.id ×ₖ κ) ∘ₘ μ := by
+    rw [Measure.comp_assoc, Kernel.parallelComp_comp_prod, Measure.compProd_eq_comp_prod,
+      Kernel.id_comp, Kernel.deterministic_comp_eq_map]
+  _ = (Kernel.id ∥ₖ Kernel.deterministic f hf) ∘ₘ (μ ⊗ₘ κ) := by rw [Measure.compProd_eq_comp_prod]
+  _ = (μ ⊗ₘ κ).map (Prod.map id f) := by
+    rw [Kernel.id, Kernel.deterministic_parallelComp_deterministic,
+      Measure.deterministic_comp_eq_map]
+
+lemma partialTraj_compProd_eq_traj_map_frestrictLe (a : ℕ) (x₀ : (i : Iic 0) → X i) :
+    (partialTraj κ 0 a x₀) ⊗ₘ (κ a) =
+      (traj κ 0 x₀).map (fun x ↦ (frestrictLe a x, x (a + 1))) := by
+  have h1 := partialTraj_compProd_traj (κ := κ) (zero_le a) x₀
+  have h2 : (fun x : Π n, X n ↦ (frestrictLe a x, x (a + 1))) =
+      (Prod.map id (fun x ↦ x (a + 1))) ∘ (fun x ↦ (frestrictLe a x, x)) := by ext <;> simp
+  rw [h2, ← Measure.map_map (by fun_prop) (by fun_prop), ← h1, ← Measure.compProd_map (by fun_prop)]
+  congr
+  have : (fun x : Π n, X n ↦ x (a + 1)) =
+      (fun x : Π i : Iic (a + 1), X i ↦ x ⟨a+1, by simp⟩) ∘ (frestrictLe (a + 1)) := by ext; simp
+  rw [this, map_comp_right _ (by fun_prop) (by fun_prop), traj_map_frestrictLe,
+    partialTraj_succ_self, ← map_comp_right _ (by fun_prop) (by fun_prop)]
+  have : (fun x : Π i : Iic (a + 1), X i ↦ x ⟨a+1, by simp⟩) ∘ IicProdIoc a (a + 1)
+      = (MeasurableEquiv.piSingleton a).symm ∘ Prod.snd := by
+    ext; simp [_root_.IicProdIoc, MeasurableEquiv.piSingleton]
+  rw [this, map_comp_right _ (by fun_prop) (by fun_prop), ← snd_eq, snd_prod,
+    ← map_comp_right _ (by fun_prop) (by fun_prop)]
+  simp
+
+lemma traj_cond_lemma1 {a : ℕ} (μ₀ : Measure ((i : Iic 0) → X i)) [IsFiniteMeasure μ₀] :
+    (traj κ 0 ∘ₘ μ₀).map (fun x ↦ (frestrictLe a x, x (a + 1)))
+      = (traj κ 0 ∘ₘ μ₀).map (frestrictLe a) ⊗ₘ κ a := by
+  rw [Measure.compProd_eq_comp_prod, Measure.map_comp _ _ (by fun_prop),
+    Measure.map_comp _ _ (by fun_prop), Measure.comp_assoc, traj_map_frestrictLe]
+  congr
+  ext x₀ : 1
+  rw [ProbabilityTheory.Kernel.comp_apply, ← Measure.compProd_eq_comp_prod]
+  symm
+  rw [Kernel.map_apply _ (by fun_prop)]
+  exact partialTraj_compProd_eq_traj_map_frestrictLe a x₀
+
+lemma condDistrib_lemma (μ₀ : Measure ((i : Iic 0) → X i)) [IsFiniteMeasure μ₀] (a : ℕ)
+    [Nonempty (X (a + 1))] [StandardBorelSpace (X (a + 1))] :
+    condDistrib (fun x ↦ x (a + 1)) (frestrictLe a) (traj κ 0 ∘ₘ μ₀)
+      =ᵐ[(traj κ 0 ∘ₘ μ₀).map (frestrictLe a)] κ a := by
+  symm
+  exact condDistrib_ae_eq_of_measure_eq_compProd (by fun_prop) (by fun_prop) _ (traj_cond_lemma1 μ₀)
+
+lemma traj_zero_map_eval_zero :
+    (Kernel.traj κ 0).map (fun h ↦ h 0)
+      = Kernel.deterministic (MeasurableEquiv.piIicZero X)
+        (MeasurableEquiv.piIicZero X).measurable := by
+  suffices (Kernel.traj κ 0).map (fun h ↦ h 0) = (Kernel.partialTraj κ 0 0).map
+      (MeasurableEquiv.piIicZero X) by
+    rwa [Kernel.partialTraj_zero,
+      Kernel.deterministic_map _ (MeasurableEquiv.piIicZero X).measurable] at this
+  rw [← Kernel.traj_map_frestrictLe, ← Kernel.map_comp_right _ (by fun_prop) (by fun_prop)]
+  congr with h
+  sorry
+
+end Traj
+
 lemma condDistrib_arm_reward [StandardBorelSpace α] [Nonempty α] [StandardBorelSpace R] [Nonempty R]
     (alg : Algorithm α R) (ν : Kernel α R) [IsMarkovKernel ν] (n : ℕ) :
     condDistrib (fun h ↦ (arm (n + 1) h, reward (n + 1) h)) (hist n) (Bandit.trajMeasure alg ν)
-      =ᵐ[(Bandit.trajMeasure alg ν).map (hist n)] Bandit.stepKernel alg ν n := by
-  sorry
+      =ᵐ[(Bandit.trajMeasure alg ν).map (hist n)] Bandit.stepKernel alg ν n :=
+  condDistrib_lemma (X := fun _ ↦ α × R)
+    ((alg.p0 ⊗ₘ ν).map (MeasurableEquiv.piIicZero (fun _ ↦ α × R)).symm)
+    (κ := Bandit.stepKernel alg ν) n
 
-lemma condDistrib_reward [StandardBorelSpace R] [Nonempty R] (alg : Algorithm α R) (ν : Kernel α R)
-    [IsMarkovKernel ν] (n : ℕ) :
+lemma condDistrib_reward [StandardBorelSpace α] [Nonempty α] [StandardBorelSpace R] [Nonempty R]
+    (alg : Algorithm α R) (ν : Kernel α R) [IsMarkovKernel ν] (n : ℕ) :
     condDistrib (reward n) (arm n) (Bandit.trajMeasure alg ν)
       =ᵐ[(Bandit.trajMeasure alg ν).map (arm n)] ν := by
-  sorry
+  cases n with
+  | zero => sorry
+  | succ n =>
+    have h_ar := condDistrib_arm_reward alg ν n
+    have h_prod := condDistrib_prod_left (X := arm (n + 1)) (Y := reward (n + 1))
+      (T := hist n) (μ := Bandit.trajMeasure alg ν) (by fun_prop) (by fun_prop) (by fun_prop)
+    sorry
 
 lemma condDistrib_arm [StandardBorelSpace α] [Nonempty α] [StandardBorelSpace R] [Nonempty R]
     (alg : Algorithm α R) (ν : Kernel α R) [IsMarkovKernel ν] (n : ℕ) :
@@ -153,6 +252,17 @@ lemma condDistrib_arm [StandardBorelSpace α] [Nonempty α] [StandardBorelSpace
       =ᵐ[(Bandit.trajMeasure alg ν).map (hist n)] alg.policy n := by
   sorry
 
+lemma hasLaw_step_zero
+    (alg : Algorithm α R) (ν : Kernel α R) [IsMarkovKernel ν] :
+    HasLaw (fun h : ℕ → α × R ↦ h 0) (alg.p0 ⊗ₘ ν) (Bandit.trajMeasure alg ν) where
+  aemeasurable := Measurable.aemeasurable (by fun_prop)
+  map_eq := by
+    simp only [Bandit.trajMeasure]
+    rw [← Measure.deterministic_comp_eq_map (by fun_prop), Measure.comp_assoc,
+      Kernel.deterministic_comp_eq_map, Bandit.traj, traj_zero_map_eval_zero,
+      Measure.deterministic_comp_eq_map, Measure.map_map (by fun_prop) (by fun_prop)]
+    simp
+
 lemma hasLaw_arm_zero [StandardBorelSpace α] [Nonempty α] [StandardBorelSpace R] [Nonempty R]
     (alg : Algorithm α R) (ν : Kernel α R) [IsMarkovKernel ν] :
     HasLaw (arm 0) alg.p0 (Bandit.trajMeasure alg ν) where
@@ -168,6 +278,37 @@ lemma condIndepFun_reward_hist_arm [StandardBorelSpace α] [Nonempty α]
   rw [condIndepFun_iff_condDistrib_prod_ae_eq_prodMkLeft (by fun_prop) (by fun_prop) (by fun_prop)]
   sorry
 
+section DetAlgorithm
+
+variable [StandardBorelSpace α] [Nonempty α] [StandardBorelSpace R] [Nonempty R]
+  {nextArm : (n : ℕ) → (Iic n → α × R) → α} {h_next : ∀ n, Measurable (nextArm n)}
+  {arm0 : α} {ν : Kernel α R} [IsMarkovKernel ν]
+
+lemma HasLaw_arm_zero_detAlgorithm :
+    HasLaw (arm 0) (Measure.dirac arm0)
+      (Bandit.trajMeasure (detAlgorithm nextArm h_next arm0) ν) where
+  map_eq := (hasLaw_arm_zero _ _).map_eq
+
+lemma arm_zero_detAlgorithm :
+    arm 0 =ᵐ[Bandit.trajMeasure (detAlgorithm nextArm h_next arm0) ν] fun _ ↦ arm0 := by
+  have h_eq : ∀ᵐ x ∂((Bandit.trajMeasure (detAlgorithm nextArm h_next arm0) ν).map (arm 0)), x
+      = arm0 := by
+    rw [(hasLaw_arm_zero _ _).map_eq]
+    simp [detAlgorithm]
+  exact ae_of_ae_map (by fun_prop) h_eq
+
+lemma arm_detAlgorithm_ae_eq (n : ℕ) :
+    arm (n + 1) =ᵐ[Bandit.trajMeasure (detAlgorithm nextArm h_next arm0) ν]
+      fun h ↦ nextArm n (fun i ↦ h i) := by
+  sorry
+
+example : ∀ᵐ h ∂(Bandit.trajMeasure (detAlgorithm nextArm h_next arm0) ν),
+    arm 0 h = arm0 ∧ ∀ n, arm (n + 1) h = nextArm n (fun i ↦ h i) := by
+  rw [eventually_and, ae_all_iff]
+  exact ⟨arm_zero_detAlgorithm, arm_detAlgorithm_ae_eq⟩
+
+end DetAlgorithm
+
 end MeasureSpace
 
 end Bandits
diff --git a/LeanBandits/ETC.lean b/LeanBandits/ETC.lean
index 6af21402..d7defd78 100644
--- a/LeanBandits/ETC.lean
+++ b/LeanBandits/ETC.lean
@@ -38,22 +38,19 @@ lemma measurable_etcNextArm (hK : 0 < K) (m n : ℕ) : Measurable (etcNextArm hK
 
 /-- The Explore-Then-Commit algorithm. -/
 noncomputable
-def etcAlgorithm (hK : 0 < K) (m : ℕ) : Algorithm (Fin K) ℝ where
-  policy n := Kernel.deterministic (etcNextArm hK m n) (by fun_prop)
-  p0 := Measure.dirac ⟨0, hK⟩
+def etcAlgorithm (hK : 0 < K) (m : ℕ) : Algorithm (Fin K) ℝ :=
+  detAlgorithm (etcNextArm hK m) (by fun_prop) ⟨0, hK⟩
 
 lemma ETC.arm_zero (hK : 0 < K) (m : ℕ) (ν : Kernel (Fin K) ℝ) [IsMarkovKernel ν] :
     arm 0 =ᵐ[Bandit.trajMeasure (etcAlgorithm hK m) ν] fun _ ↦ ⟨0, hK⟩ := by
-  have h_eq : ∀ᵐ x ∂((Bandit.trajMeasure (etcAlgorithm hK m) ν).map (arm 0)), x = ⟨0, hK⟩ := by
-    have : Nonempty (Fin K) := Fin.pos_iff_nonempty.mp hK
-    rw [(hasLaw_arm_zero _ _).map_eq]
-    simp [etcAlgorithm]
-  exact ae_of_ae_map (by fun_prop) h_eq
+  have : Nonempty (Fin K) := Fin.pos_iff_nonempty.mp hK
+  exact arm_zero_detAlgorithm
 
 lemma ETC.arm_ae_eq_etcNextArm (hK : 0 < K) (m : ℕ) (ν : Kernel (Fin K) ℝ) [IsMarkovKernel ν]
     (n : ℕ) :
     arm (n + 1) =ᵐ[(Bandit.trajMeasure (etcAlgorithm hK m) ν)]
       fun h ↦ etcNextArm hK m n (fun i ↦ h i) := by
-  sorry
+  have : Nonempty (Fin K) := Fin.pos_iff_nonempty.mp hK
+  exact arm_detAlgorithm_ae_eq n
 
 end Bandits
diff --git a/LeanBandits/ForMathlib/CondDistrib.lean b/LeanBandits/ForMathlib/CondDistrib.lean
index 00ddff42..0e069f4e 100644
--- a/LeanBandits/ForMathlib/CondDistrib.lean
+++ b/LeanBandits/ForMathlib/CondDistrib.lean
@@ -68,6 +68,14 @@ end MeasureTheory.Measure
 
 namespace ProbabilityTheory
 
+lemma Kernel.deterministic_parallelComp_deterministic
+    {f : α → γ} {g : β → δ} (hf : Measurable f) (hg : Measurable g) :
+    (deterministic f hf) ∥ₖ (deterministic g hg)
+      = deterministic (Prod.map f g) (hf.prodMap hg) := by
+  ext x : 1
+  rw [parallelComp_apply, deterministic_apply, deterministic_apply, deterministic_apply, Prod.map,
+    Measure.dirac_prod_dirac]
+
 lemma Kernel.prod_apply_prod {κ : Kernel α β} {η : Kernel α γ}
     [IsSFiniteKernel κ] [IsSFiniteKernel η] {s : Set β} {t : Set γ} {a : α} :
     (κ ×ₖ η) a (s ×ˢ t) = (κ a s) * (η a t) := by
@@ -513,6 +521,26 @@ lemma condDistrib_fst_prod (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ)
         fun_prop
   · fun_prop
 
+lemma Measure.compProd_assoc {μ : Measure α} {κ : Kernel α β} {η : Kernel (α × β) γ}
+    [SFinite μ] [IsSFiniteKernel κ] [IsSFiniteKernel η] :
+    (μ ⊗ₘ κ) ⊗ₘ η = (μ ⊗ₘ (κ ⊗ₖ η)).map MeasurableEquiv.prodAssoc.symm := by
+  sorry
+
+lemma Measure.compProd_assoc' {μ : Measure α} {κ : Kernel α β} {η : Kernel (α × β) γ}
+    [SFinite μ] [IsSFiniteKernel κ] [IsSFiniteKernel η] :
+    μ ⊗ₘ (κ ⊗ₖ η) = ((μ ⊗ₘ κ) ⊗ₘ η).map MeasurableEquiv.prodAssoc := by
+  simp [Measure.compProd_assoc]
+
+lemma condDistrib_prod_left [StandardBorelSpace β] [Nonempty β]
+    (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ) (hT : AEMeasurable T μ) :
+    condDistrib (fun ω ↦ (X ω, Y ω)) T μ
+      =ᵐ[μ.map T] condDistrib X T μ ⊗ₖ condDistrib Y (fun ω ↦ (T ω, X ω)) μ := by
+  refine condDistrib_ae_eq_of_measure_eq_compProd₀ (μ := μ) hT (by fun_prop)
+    (condDistrib X T μ ⊗ₖ condDistrib Y (fun ω ↦ (T ω, X ω)) μ) ?_
+  rw [Measure.compProd_assoc', compProd_map_condDistrib hX, compProd_map_condDistrib hY,
+    AEMeasurable.map_map_of_aemeasurable (by fun_prop) (by fun_prop)]
+  rfl
+
 end CondDistrib
 
 section Cond
diff --git a/LeanBandits/RewardByCountMeasure.lean b/LeanBandits/RewardByCountMeasure.lean
index 99224810..cb581c7b 100644
--- a/LeanBandits/RewardByCountMeasure.lean
+++ b/LeanBandits/RewardByCountMeasure.lean
@@ -102,7 +102,7 @@ notation "𝓛[" Y " | " X " ← " x "; " μ "]" => Measure.map Y (μ[|X ⁻¹'
 notation "𝓛[" Y " | " X "; " μ "]" => condDistrib Y X μ
 
 omit [DecidableEq α] [MeasurableSingletonClass α] in
-lemma condDistrib_reward' (n : ℕ) :
+lemma condDistrib_reward' [StandardBorelSpace α] [Nonempty α] (n : ℕ) :
     𝓛[fun ω ↦ reward n ω.1 | fun ω ↦ arm n ω.1; Bandit.measure alg ν]
       =ᵐ[(Bandit.measure alg ν).map (fun ω ↦ arm n ω.1)] ν := by
   let μ := Bandit.measure alg ν
@@ -122,7 +122,7 @@ lemma condDistrib_reward' (n : ℕ) :
   rw [h_prod, h_eq]
 
 omit [DecidableEq α] in
-lemma reward_cond_arm [Countable α] (a : α) (n : ℕ)
+lemma reward_cond_arm [StandardBorelSpace α] [Nonempty α] [Countable α] (a : α) (n : ℕ)
     (hμa : (Bandit.measure alg ν).map (fun ω ↦ arm n ω.1) {a} ≠ 0) :
     𝓛[fun ω ↦ reward n ω.1 | fun ω ↦ arm n ω.1 ← a; Bandit.measure alg ν] = ν a := by
   let μ := Bandit.measure alg ν
diff --git a/LeanBandits/UCB.lean b/LeanBandits/UCB.lean
index d188a6c3..3ee285ef 100644
--- a/LeanBandits/UCB.lean
+++ b/LeanBandits/UCB.lean
@@ -43,9 +43,8 @@ lemma measurable_ucbNextArm (c : ℝ) (n : ℕ) : Measurable (ucbNextArm c n (α
 
 /-- The UCB algorithm. -/
 noncomputable
-def ucbAlgorithm (c : ℝ) : Algorithm α ℝ where
-  policy n := Kernel.deterministic (ucbNextArm c n) (by fun_prop)
-  p0 := Measure.dirac (Classical.arbitrary α)
+def ucbAlgorithm (c : ℝ) : Algorithm α ℝ :=
+  detAlgorithm (ucbNextArm c) (by fun_prop) (Classical.arbitrary α)
 
 end Algorithm