/-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
    {f : X → Y → E} (hf : Continuous f.uncurry) (h'f : HasCompactSupport f.uncurry)
    {μ : Measure X} {ν : Measure Y} [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts ν] :
    ∫ x, (∫ y, f x y ∂ν) ∂μ = ∫ y, (∫ x, f x y ∂μ) ∂ν := by
  let U := Prod.fst '' (tsupport f.uncurry)
  have : Fact (μ U < ∞) := ⟨(IsCompact.image h'f continuous_fst).measure_lt_top⟩
  let V := Prod.snd '' (tsupport f.uncurry)
  have : Fact (ν V < ∞) := ⟨(IsCompact.image h'f continuous_snd).measure_lt_top⟩
  calc
  ∫ x, (∫ y, f x y ∂ν) ∂μ = ∫ x, (∫ y in V, f x y ∂ν) ∂μ := by
    congr 1 with x
    apply (set_integral_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 (set_integral_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 set_integral_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
    

Proof state:
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

Next tactic:
apply set_integral_eq_integral_of_forall_compl_eq_zero (fun x hx ↦ ?_)