Theorem: MeasureTheory.integral_integral_swap_of_hasCompactSupport
File: Mathlib/MeasureTheory/Constructions/Prod/Integral.lean

tactic: let U := Prod.fst '' (tsupport f.uncurry)
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
⊢ ∫ (x : X), ∫ (y : Y), f x y ∂ν ∂μ = ∫ (y : Y), ∫ (x : X), f x y ∂μ ∂ν
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
⊢ ∫ (x : X), ∫ (y : Y), f x y ∂ν ∂μ = ∫ (y : Y), ∫ (x : X), f x y ∂μ ∂ν

tactic: have : Fact (μ U < ∞) := ⟨(IsCompact.image h'f continuous_fst).measure_lt_top⟩
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
⊢ ∫ (x : X), ∫ (y : Y), f x y ∂ν ∂μ = ∫ (y : Y), ∫ (x : X), f x y ∂μ ∂ν
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this : Fact (μ U < ⊤)
⊢ ∫ (x : X), ∫ (y : Y), f x y ∂ν ∂μ = ∫ (y : Y), ∫ (x : X), f x y ∂μ ∂ν

tactic: let V := Prod.snd '' (tsupport f.uncurry)
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this : Fact (μ U < ⊤)
⊢ ∫ (x : X), ∫ (y : Y), f x y ∂ν ∂μ = ∫ (y : Y), ∫ (x : X), f x y ∂μ ∂ν
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
⊢ ∫ (x : X), ∫ (y : Y), f x y ∂ν ∂μ = ∫ (y : Y), ∫ (x : X), f x y ∂μ ∂ν

tactic: have : Fact (ν V < ∞) := ⟨(IsCompact.image h'f continuous_snd).measure_lt_top⟩
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
⊢ ∫ (x : X), ∫ (y : Y), f x y ∂ν ∂μ = ∫ (y : Y), ∫ (x : X), f x y ∂μ ∂ν
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
⊢ ∫ (x : X), ∫ (y : Y), f x y ∂ν ∂μ = ∫ (y : Y), ∫ (x : X), f x y ∂μ ∂ν

tactic: calc
∫ x, (∫ y, f x y ∂ν) ∂μ = ∫ x, (∫ y in V, f x y ∂ν) ∂μ := by
  congr 1 with x
  apply (setIntegral_eq_integral_of_forall_compl_eq_zero (fun y hy ↦ ?_)).symm
  contrapose! hy
  have : (x, y) ∈ Function.support f.uncurry := hy
  exact mem_image_of_mem _ (subset_tsupport _ this)
_ = ∫ x in U, (∫ y in V, f x y ∂ν) ∂μ := by
  apply (setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx ↦ ?_)).symm
  have : ∀ y, f x y = 0 := by
    intro y
    contrapose! hx
    have : (x, y) ∈ Function.support f.uncurry := hx
    exact mem_image_of_mem _ (subset_tsupport _ this)
  simp [this]
_ = ∫ y in V, (∫ x in U, f x y ∂μ) ∂ν := by
  apply integral_integral_swap
  apply (integrableOn_iff_integrable_of_support_subset (subset_tsupport f.uncurry)).mp
  refine ⟨(h'f.stronglyMeasurable_of_prod hf).aestronglyMeasurable, ?_⟩
  obtain ⟨C, hC⟩ : ∃ C, ∀ p, ‖f.uncurry p‖ ≤ C := hf.bounded_above_of_compact_support h'f
  exact hasFiniteIntegral_of_bounded (C := C) (eventually_of_forall hC)
_ = ∫ y, (∫ x in U, f x y ∂μ) ∂ν := by
  apply setIntegral_eq_integral_of_forall_compl_eq_zero (fun y hy ↦ ?_)
  have : ∀ x, f x y = 0 := by
    intro x
    contrapose! hy
    have : (x, y) ∈ Function.support f.uncurry := hy
    exact mem_image_of_mem _ (subset_tsupport _ this)
  simp [this]
_ = ∫ y, (∫ x, f x y ∂μ) ∂ν := by
  congr 1 with y
  apply setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx ↦ ?_)
  contrapose! hx
  have : (x, y) ∈ Function.support f.uncurry := hx
  exact mem_image_of_mem _ (subset_tsupport _ this)
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
⊢ ∫ (x : X), ∫ (y : Y), f x y ∂ν ∂μ = ∫ (y : Y), ∫ (x : X), f x y ∂μ ∂ν
state_after: no goals

tactic: congr 1 with x
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
⊢ ∫ (x : X), ∫ (y : Y), f x y ∂ν ∂μ = ∫ (x : X), ∫ (y : Y) in V, f x y ∂ν ∂μ
state_after: case e_f.h
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
x : X
⊢ ∫ (y : Y), f x y ∂ν = ∫ (y : Y) in V, f x y ∂ν

tactic: apply (setIntegral_eq_integral_of_forall_compl_eq_zero (fun y hy ↦ ?_)).symm
state_before: case e_f.h
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
x : X
⊢ ∫ (y : Y), f x y ∂ν = ∫ (y : Y) in V, f x y ∂ν
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
x : X
y : Y
hy : y ∉ V
⊢ f x y = 0

tactic: contrapose! hy
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
x : X
y : Y
hy : y ∉ V
⊢ f x y = 0
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
x : X
y : Y
hy : f x y ≠ 0
⊢ y ∈ V

tactic: have : (x, y) ∈ Function.support f.uncurry := hy
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
x : X
y : Y
hy : f x y ≠ 0
⊢ y ∈ V
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝¹ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this✝ : Fact (ν V < ⊤)
x : X
y : Y
hy : f x y ≠ 0
this : (x, y) ∈ support (uncurry f)
⊢ y ∈ V

tactic: exact mem_image_of_mem _ (subset_tsupport _ this)
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝¹ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this✝ : Fact (ν V < ⊤)
x : X
y : Y
hy : f x y ≠ 0
this : (x, y) ∈ support (uncurry f)
⊢ y ∈ V
state_after: no goals

tactic: apply (setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx ↦ ?_)).symm
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
⊢ ∫ (x : X), ∫ (y : Y) in V, f x y ∂ν ∂μ = ∫ (x : X) in U, ∫ (y : Y) in V, f x y ∂ν ∂μ
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
x : X
hx : x ∉ U
⊢ ∫ (y : Y) in V, f x y ∂ν = 0

tactic: have : ∀ y, f x y = 0 := by
  intro y
  contrapose! hx
  have : (x, y) ∈ Function.support f.uncurry := hx
  exact mem_image_of_mem _ (subset_tsupport _ this)
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
x : X
hx : x ∉ U
⊢ ∫ (y : Y) in V, f x y ∂ν = 0
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝¹ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this✝ : Fact (ν V < ⊤)
x : X
hx : x ∉ U
this : ∀ (y : Y), f x y = 0
⊢ ∫ (y : Y) in V, f x y ∂ν = 0

tactic: simp [this]
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝¹ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this✝ : Fact (ν V < ⊤)
x : X
hx : x ∉ U
this : ∀ (y : Y), f x y = 0
⊢ ∫ (y : Y) in V, f x y ∂ν = 0
state_after: no goals

tactic: intro y
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
x : X
hx : x ∉ U
⊢ ∀ (y : Y), f x y = 0
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
x : X
hx : x ∉ U
y : Y
⊢ f x y = 0

tactic: contrapose! hx
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
x : X
hx : x ∉ U
y : Y
⊢ f x y = 0
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
x : X
y : Y
hx : f x y ≠ 0
⊢ x ∈ U

tactic: have : (x, y) ∈ Function.support f.uncurry := hx
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
x : X
y : Y
hx : f x y ≠ 0
⊢ x ∈ U
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝¹ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this✝ : Fact (ν V < ⊤)
x : X
y : Y
hx : f x y ≠ 0
this : (x, y) ∈ support (uncurry f)
⊢ x ∈ U

tactic: exact mem_image_of_mem _ (subset_tsupport _ this)
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝¹ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this✝ : Fact (ν V < ⊤)
x : X
y : Y
hx : f x y ≠ 0
this : (x, y) ∈ support (uncurry f)
⊢ x ∈ U
state_after: no goals

tactic: apply integral_integral_swap
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
⊢ ∫ (x : X) in U, ∫ (y : Y) in V, f x y ∂ν ∂μ = ∫ (y : Y) in V, ∫ (x : X) in U, f x y ∂μ ∂ν
state_after: case hf
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
⊢ Integrable (uncurry f) ((μ.restrict U).prod (ν.restrict V))

tactic: apply (integrableOn_iff_integrable_of_support_subset (subset_tsupport f.uncurry)).mp
state_before: case hf
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
⊢ Integrable (uncurry f) ((μ.restrict U).prod (ν.restrict V))
state_after: case hf
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
⊢ IntegrableOn (uncurry f) (tsupport (uncurry f)) ((μ.restrict U).prod (ν.restrict V))

tactic: refine ⟨(h'f.stronglyMeasurable_of_prod hf).aestronglyMeasurable, ?_⟩
state_before: case hf
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
⊢ IntegrableOn (uncurry f) (tsupport (uncurry f)) ((μ.restrict U).prod (ν.restrict V))
state_after: case hf
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
⊢ HasFiniteIntegral (uncurry f) (((μ.restrict U).prod (ν.restrict V)).restrict (tsupport (uncurry f)))

tactic: obtain ⟨C, hC⟩ : ∃ C, ∀ p, ‖f.uncurry p‖ ≤ C := hf.bounded_above_of_compact_support h'f
state_before: case hf
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
⊢ HasFiniteIntegral (uncurry f) (((μ.restrict U).prod (ν.restrict V)).restrict (tsupport (uncurry f)))
state_after: case hf.intro
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
C : ℝ
hC : ∀ (p : X × Y), ‖uncurry f p‖ ≤ C
⊢ HasFiniteIntegral (uncurry f) (((μ.restrict U).prod (ν.restrict V)).restrict (tsupport (uncurry f)))

tactic: exact hasFiniteIntegral_of_bounded (C := C) (eventually_of_forall hC)
state_before: case hf.intro
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
C : ℝ
hC : ∀ (p : X × Y), ‖uncurry f p‖ ≤ C
⊢ HasFiniteIntegral (uncurry f) (((μ.restrict U).prod (ν.restrict V)).restrict (tsupport (uncurry f)))
state_after: no goals

tactic: apply setIntegral_eq_integral_of_forall_compl_eq_zero (fun y hy ↦ ?_)
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
⊢ ∫ (y : Y) in V, ∫ (x : X) in U, f x y ∂μ ∂ν = ∫ (y : Y), ∫ (x : X) in U, f x y ∂μ ∂ν
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
y : Y
hy : y ∉ V
⊢ ∫ (x : X) in U, f x y ∂μ = 0

tactic: have : ∀ x, f x y = 0 := by
  intro x
  contrapose! hy
  have : (x, y) ∈ Function.support f.uncurry := hy
  exact mem_image_of_mem _ (subset_tsupport _ this)
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
y : Y
hy : y ∉ V
⊢ ∫ (x : X) in U, f x y ∂μ = 0
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝¹ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this✝ : Fact (ν V < ⊤)
y : Y
hy : y ∉ V
this : ∀ (x : X), f x y = 0
⊢ ∫ (x : X) in U, f x y ∂μ = 0

tactic: simp [this]
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝¹ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this✝ : Fact (ν V < ⊤)
y : Y
hy : y ∉ V
this : ∀ (x : X), f x y = 0
⊢ ∫ (x : X) in U, f x y ∂μ = 0
state_after: no goals

tactic: intro x
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
y : Y
hy : y ∉ V
⊢ ∀ (x : X), f x y = 0
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
y : Y
hy : y ∉ V
x : X
⊢ f x y = 0

tactic: contrapose! hy
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
y : Y
hy : y ∉ V
x : X
⊢ f x y = 0
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
y : Y
x : X
hy : f x y ≠ 0
⊢ y ∈ V

tactic: have : (x, y) ∈ Function.support f.uncurry := hy
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
y : Y
x : X
hy : f x y ≠ 0
⊢ y ∈ V
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝¹ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this✝ : Fact (ν V < ⊤)
y : Y
x : X
hy : f x y ≠ 0
this : (x, y) ∈ support (uncurry f)
⊢ y ∈ V

tactic: exact mem_image_of_mem _ (subset_tsupport _ this)
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝¹ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this✝ : Fact (ν V < ⊤)
y : Y
x : X
hy : f x y ≠ 0
this : (x, y) ∈ support (uncurry f)
⊢ y ∈ V
state_after: no goals

tactic: congr 1 with y
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
⊢ ∫ (y : Y), ∫ (x : X) in U, f x y ∂μ ∂ν = ∫ (y : Y), ∫ (x : X), f x y ∂μ ∂ν
state_after: case e_f.h
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
y : Y
⊢ ∫ (x : X) in U, f x y ∂μ = ∫ (x : X), f x y ∂μ

tactic: apply setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx ↦ ?_)
state_before: case e_f.h
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
y : Y
⊢ ∫ (x : X) in U, f x y ∂μ = ∫ (x : X), f x y ∂μ
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
y : Y
x : X
hx : x ∉ U
⊢ f x y = 0

tactic: contrapose! hx
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
y : Y
x : X
hx : x ∉ U
⊢ f x y = 0
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
y : Y
x : X
hx : f x y ≠ 0
⊢ x ∈ U

tactic: have : (x, y) ∈ Function.support f.uncurry := hx
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
y : Y
x : X
hx : f x y ≠ 0
⊢ x ∈ U
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝¹ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this✝ : Fact (ν V < ⊤)
y : Y
x : X
hx : f x y ≠ 0
this : (x, y) ∈ support (uncurry f)
⊢ x ∈ U

tactic: exact mem_image_of_mem _ (subset_tsupport _ this)
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFinite ν✝
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : SigmaFinite μ✝
E' : Type u_7
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace ℝ E'
X : Type u_8
Y : Type u_9
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝¹ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this✝ : Fact (ν V < ⊤)
y : Y
x : X
hx : f x y ≠ 0
this : (x, y) ∈ support (uncurry f)
⊢ x ∈ U
state_after: no goals