Theorem: ProbabilityTheory.kernel.iIndepFun_iff_measure_inter_preimage_eq_mul
File: Mathlib/Probability/Independence/Kernel.lean

tactic: refine ⟨fun h S sets h_meas => h _ fun i hi_mem => ⟨sets i, h_meas i hi_mem, rfl⟩, ?_⟩
state_before: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
⊢ iIndepFun m f κ μ ↔
    ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
      (∀ i ∈ S, MeasurableSet (sets i)) →
        ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
state_after: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
⊢ (∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
      (∀ i ∈ S, MeasurableSet (sets i)) →
        ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)) →
    iIndepFun m f κ μ

tactic: intro h S setsΩ h_meas
state_before: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
⊢ (∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
      (∀ i ∈ S, MeasurableSet (sets i)) →
        ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)) →
    iIndepFun m f κ μ
state_after: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, setsΩ i) = ∏ i ∈ S, (κ a) (setsΩ i)

tactic: let setsβ : ∀ i : ι, Set (β i) := fun i =>
  dite (i ∈ S) (fun hi_mem => (h_meas i hi_mem).choose) fun _ => Set.univ
state_before: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, setsΩ i) = ∏ i ∈ S, (κ a) (setsΩ i)
state_after: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, setsΩ i) = ∏ i ∈ S, (κ a) (setsΩ i)

tactic: have h_measβ : ∀ i ∈ S, MeasurableSet[m i] (setsβ i) := by
  intro i hi_mem
  simp_rw [setsβ, dif_pos hi_mem]
  exact (h_meas i hi_mem).choose_spec.1
state_before: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, setsΩ i) = ∏ i ∈ S, (κ a) (setsΩ i)
state_after: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, setsΩ i) = ∏ i ∈ S, (κ a) (setsΩ i)

tactic: have h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i := by
  intro i hi_mem
  simp_rw [setsβ, dif_pos hi_mem]
  exact (h_meas i hi_mem).choose_spec.2.symm
state_before: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, setsΩ i) = ∏ i ∈ S, (κ a) (setsΩ i)
state_after: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, setsΩ i) = ∏ i ∈ S, (κ a) (setsΩ i)

tactic: have h_right_eq : ∀ a, (∏ i ∈ S, κ a (setsΩ i)) = ∏ i ∈ S, κ a ((f i) ⁻¹' (setsβ i)) := by
  refine fun a ↦ Finset.prod_congr rfl fun i hi_mem => ?_
  rw [h_preim i hi_mem]
state_before: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
h_left_eq : ∀ (a : α), (κ a) (⋂ i ∈ S, setsΩ i) = (κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i)
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, setsΩ i) = ∏ i ∈ S, (κ a) (setsΩ i)
state_after: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
h_left_eq : ∀ (a : α), (κ a) (⋂ i ∈ S, setsΩ i) = (κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i)
h_right_eq : ∀ (a : α), ∏ i ∈ S, (κ a) (setsΩ i) = ∏ i ∈ S, (κ a) (f i ⁻¹' setsβ i)
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, setsΩ i) = ∏ i ∈ S, (κ a) (setsΩ i)

tactic: filter_upwards [h S h_measβ] with a ha
state_before: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
h_left_eq : ∀ (a : α), (κ a) (⋂ i ∈ S, setsΩ i) = (κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i)
h_right_eq : ∀ (a : α), ∏ i ∈ S, (κ a) (setsΩ i) = ∏ i ∈ S, (κ a) (f i ⁻¹' setsβ i)
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, setsΩ i) = ∏ i ∈ S, (κ a) (setsΩ i)
state_after: case h
α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
h_left_eq : ∀ (a : α), (κ a) (⋂ i ∈ S, setsΩ i) = (κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i)
h_right_eq : ∀ (a : α), ∏ i ∈ S, (κ a) (setsΩ i) = ∏ i ∈ S, (κ a) (f i ⁻¹' setsβ i)
a : α
ha : (κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) = ∏ i ∈ S, (κ a) (f i ⁻¹' setsβ i)
⊢ (κ a) (⋂ i ∈ S, setsΩ i) = ∏ i ∈ S, (κ a) (setsΩ i)

tactic: rw [h_left_eq a, h_right_eq a, ha]
state_before: case h
α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
h_left_eq : ∀ (a : α), (κ a) (⋂ i ∈ S, setsΩ i) = (κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i)
h_right_eq : ∀ (a : α), ∏ i ∈ S, (κ a) (setsΩ i) = ∏ i ∈ S, (κ a) (f i ⁻¹' setsβ i)
a : α
ha : (κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) = ∏ i ∈ S, (κ a) (f i ⁻¹' setsβ i)
⊢ (κ a) (⋂ i ∈ S, setsΩ i) = ∏ i ∈ S, (κ a) (setsΩ i)
state_after: no goals

tactic: intro i hi_mem
state_before: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
⊢ ∀ i ∈ S, MeasurableSet (setsβ i)
state_after: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
i : ι
hi_mem : i ∈ S
⊢ MeasurableSet (setsβ i)

tactic: simp_rw [setsβ, dif_pos hi_mem]
state_before: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
i : ι
hi_mem : i ∈ S
⊢ MeasurableSet (setsβ i)
state_after: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
i : ι
hi_mem : i ∈ S
⊢ MeasurableSet (Exists.choose ⋯)

tactic: exact (h_meas i hi_mem).choose_spec.1
state_before: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
i : ι
hi_mem : i ∈ S
⊢ MeasurableSet (Exists.choose ⋯)
state_after: no goals

tactic: intro i hi_mem
state_before: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
⊢ ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
state_after: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
i : ι
hi_mem : i ∈ S
⊢ setsΩ i = f i ⁻¹' setsβ i

tactic: simp_rw [setsβ, dif_pos hi_mem]
state_before: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
i : ι
hi_mem : i ∈ S
⊢ setsΩ i = f i ⁻¹' setsβ i
state_after: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
i : ι
hi_mem : i ∈ S
⊢ setsΩ i = f i ⁻¹' Exists.choose ⋯

tactic: exact (h_meas i hi_mem).choose_spec.2.symm
state_before: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
i : ι
hi_mem : i ∈ S
⊢ setsΩ i = f i ⁻¹' Exists.choose ⋯
state_after: no goals

tactic: intro a
state_before: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
⊢ ∀ (a : α), (κ a) (⋂ i ∈ S, setsΩ i) = (κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i)
state_after: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
a : α
⊢ (κ a) (⋂ i ∈ S, setsΩ i) = (κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i)

tactic: congr with x
state_before: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
a : α
⊢ (κ a) (⋂ i ∈ S, setsΩ i) = (κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i)
state_after: case h.e_6.h.h
α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
a : α
x : Ω
⊢ x ∈ ⋂ i ∈ S, setsΩ i ↔ x ∈ ⋂ i ∈ S, f i ⁻¹' setsβ i

tactic: simp_rw [Set.mem_iInter]
state_before: case h.e_6.h.h
α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
a : α
x : Ω
⊢ x ∈ ⋂ i ∈ S, setsΩ i ↔ x ∈ ⋂ i ∈ S, f i ⁻¹' setsβ i
state_after: case h.e_6.h.h
α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
a : α
x : Ω
⊢ (∀ i ∈ S, x ∈ setsΩ i) ↔ ∀ i ∈ S, x ∈ f i ⁻¹' setsβ i

tactic: constructor <;> intro h i hi_mem <;> specialize h i hi_mem
state_before: case h.e_6.h.h
α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
a : α
x : Ω
⊢ (∀ i ∈ S, x ∈ setsΩ i) ↔ ∀ i ∈ S, x ∈ f i ⁻¹' setsβ i
state_after: case h.e_6.h.h.mp
α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h✝ :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
a : α
x : Ω
i : ι
hi_mem : i ∈ S
h : x ∈ setsΩ i
⊢ x ∈ f i ⁻¹' setsβ i

case h.e_6.h.h.mpr
α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h✝ :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
a : α
x : Ω
i : ι
hi_mem : i ∈ S
h : x ∈ f i ⁻¹' setsβ i
⊢ x ∈ setsΩ i

tactic: rwa [h_preim i hi_mem] at h
state_before: case h.e_6.h.h.mp
α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h✝ :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
a : α
x : Ω
i : ι
hi_mem : i ∈ S
h : x ∈ setsΩ i
⊢ x ∈ f i ⁻¹' setsβ i
state_after: no goals

tactic: rwa [h_preim i hi_mem]
state_before: case h.e_6.h.h.mpr
α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h✝ :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
a : α
x : Ω
i : ι
hi_mem : i ∈ S
h : x ∈ f i ⁻¹' setsβ i
⊢ x ∈ setsΩ i
state_after: no goals

tactic: refine fun a ↦ Finset.prod_congr rfl fun i hi_mem => ?_
state_before: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
h_left_eq : ∀ (a : α), (κ a) (⋂ i ∈ S, setsΩ i) = (κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i)
⊢ ∀ (a : α), ∏ i ∈ S, (κ a) (setsΩ i) = ∏ i ∈ S, (κ a) (f i ⁻¹' setsβ i)
state_after: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
h_left_eq : ∀ (a : α), (κ a) (⋂ i ∈ S, setsΩ i) = (κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i)
a : α
i : ι
hi_mem : i ∈ S
⊢ (κ a) (setsΩ i) = (κ a) (f i ⁻¹' setsβ i)

tactic: rw [h_preim i hi_mem]
state_before: α : Type u_1
Ω : Type u_2
ι✝ : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
S : Finset ι
setsΩ : ι → Set Ω
h_meas : ∀ i ∈ S, setsΩ i ∈ (fun x => {s | MeasurableSet s}) i
setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose ⋯ else Set.univ
h_measβ : ∀ i ∈ S, MeasurableSet (setsβ i)
h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i
h_left_eq : ∀ (a : α), (κ a) (⋂ i ∈ S, setsΩ i) = (κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i)
a : α
i : ι
hi_mem : i ∈ S
⊢ (κ a) (setsΩ i) = (κ a) (f i ⁻¹' setsβ i)
state_after: no goals