theorem lintegral_condKernelReal_mem {s : Set (α × ℝ)} (hs : MeasurableSet s) :
    ∫⁻ a, condKernelReal ρ a {x | (a, x) ∈ s} ∂ρ.fst = ρ s := by
  -- `set_lintegral_condKernelReal_prod` gives the result for sets of the form `t₁ × t₂`. These
  -- sets form a π-system that generates the product σ-algebra, hence we can get the same equality
  -- for any measurable set `s`.
  apply MeasurableSpace.induction_on_inter generateFrom_prod.symm isPiSystem_prod _ _ _ _ hs
  · simp only [mem_empty_iff_false, setOf_false, measure_empty, lintegral_const,
      zero_mul]
  · intro t ht
    rw [mem_image2] at ht
    obtain ⟨t₁, t₂, ht₁, ht₂, rfl⟩ := ht
    have h_prod_eq_snd : ∀ a ∈ t₁, {x : ℝ | (a, x) ∈ t₁ ×ˢ t₂} = t₂ := by
      intro a ha
      simp only [ha, prod_mk_mem_set_prod_eq, true_and_iff, setOf_mem_eq]
    rw [← lintegral_add_compl _ ht₁]
    have h_eq1 : ∫⁻ a in t₁, condKernelReal ρ a {x : ℝ | (a, x) ∈ t₁ ×ˢ t₂} ∂ρ.fst =
        ∫⁻ a in t₁, condKernelReal ρ a t₂ ∂ρ.fst := by
      refine' set_lintegral_congr_fun ht₁ (eventually_of_forall fun a ha => _)
      rw [h_prod_eq_snd a ha]
    have h_eq2 : ∫⁻ a in t₁ᶜ, condKernelReal ρ a {x : ℝ | (a, x) ∈ t₁ ×ˢ t₂} ∂ρ.fst = 0 := by
      suffices h_eq_zero : ∀ a ∈ t₁ᶜ, condKernelReal ρ a {x : ℝ | (a, x) ∈ t₁ ×ˢ t₂} = 0
      · rw [set_lintegral_congr_fun ht₁.compl (eventually_of_forall h_eq_zero)]
        simp only [lintegral_const, zero_mul]
      intro a hat₁
      rw [mem_compl_iff] at hat₁
      simp only [hat₁, prod_mk_mem_set_prod_eq, false_and_iff, setOf_false, measure_empty]
    rw [h_eq1, h_eq2, add_zero]
    exact set_lintegral_condKernelReal_prod ρ ht₁ ht₂
  · intro t ht ht_eq
    calc
      ∫⁻ a, condKernelReal ρ a {x : ℝ | (a, x) ∈ tᶜ} ∂ρ.fst =
          ∫⁻ a, condKernelReal ρ a {x : ℝ | (a, x) ∈ t}ᶜ ∂ρ.fst := rfl
      _ = ∫⁻ a, condKernelReal ρ a univ - condKernelReal ρ a {x : ℝ | (a, x) ∈ t} ∂ρ.fst := by
        congr with a : 1
        exact measure_compl (measurable_prod_mk_left ht) (measure_ne_top (condKernelReal ρ a) _)
      _ = ∫⁻ a, condKernelReal ρ a univ ∂ρ.fst -
          ∫⁻ a, condKernelReal ρ a {x : ℝ | (a, x) ∈ t} ∂ρ.fst := by
        have h_le : (fun a => condKernelReal ρ a {x : ℝ | (a, x) ∈ t}) ≤ᵐ[ρ.fst] fun a =>
            condKernelReal ρ a univ := eventually_of_forall fun a => measure_mono (subset_univ _)
        rw [lintegral_sub _ _ h_le]
        · exact kernel.measurable_kernel_prod_mk_left ht
        refine' ((lintegral_mono_ae h_le).trans_lt _).ne
        rw [lintegral_condKernelReal_univ]
        exact measure_lt_top ρ univ
      _ = ρ univ - ρ t := by rw [ht_eq, lintegral_condKernelReal_univ]
      _ = ρ tᶜ := (measure_compl ht (measure_ne_top _ _)).symm
  · intro f hf_disj hf_meas hf_eq
    have h_eq : ∀ a, {x | (a, x) ∈ ⋃ i, f i} = ⋃ i, {x | (a, x) ∈ f i} := by
      intro a
      ext1 x
      simp only [mem_iUnion, mem_setOf_eq]
    simp_rw [h_eq]
    have h_disj : ∀ a, Pairwise (Disjoint on fun i => {x | (a, x) ∈ f i}) := by
      intro a i j hij
      have h_disj := hf_disj hij
      rw [Function.onFun, disjoint_iff_inter_eq_empty] at h_disj ⊢
      ext1 x
      simp only [mem_inter_iff, mem_setOf_eq, mem_empty_iff_false, iff_false_iff]
      intro h_mem_both
      suffices (a, x) ∈ ∅ by rwa [mem_empty_iff_false] at this
      rwa [← h_disj, mem_inter_iff]
    

Proof state:
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set (α × ℝ)
hs : MeasurableSet s
f : ℕ → Set (α × ℝ)
hf_disj : Pairwise (Disjoint on f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf_eq : ∀ (i : ℕ), ∫⁻ (a : α), ↑↑((condKernelReal ρ) a) {x | (a, x) ∈ f i} ∂Measure.fst ρ = ↑↑ρ (f i)
h_eq : ∀ (a : α), {x | (a, x) ∈ ⋃ i, f i} = ⋃ i, {x | (a, x) ∈ f i}
h_disj : ∀ (a : α), Pairwise (Disjoint on fun i => {x | (a, x) ∈ f i})
⊢ ∫⁻ (a : α), ↑↑((condKernelReal ρ) a) (⋃ i, {x | (a, x) ∈ f i}) ∂Measure.fst ρ = ↑↑ρ (⋃ i, f i)

Next tactic:
calc
      ∫⁻ a, condKernelReal ρ a (⋃ i, {x | (a, x) ∈ f i}) ∂ρ.fst =
          ∫⁻ a, ∑' i, condKernelReal ρ a {x | (a, x) ∈ f i} ∂ρ.fst := by
        congr with a : 1
        rw [measure_iUnion (h_disj a) fun i => measurable_prod_mk_left (hf_meas i)]
      _ = ∑' i, ∫⁻ a, condKernelReal ρ a {x | (a, x) ∈ f i} ∂ρ.fst :=
        (lintegral_tsum fun i => (kernel.measurable_kernel_prod_mk_left (hf_meas i)).aemeasurable)
      _ = ∑' i, ρ (f i) := by simp_rw [hf_eq]
      _ = ρ (iUnion f) := (measure_iUnion hf_disj hf_meas).symm