/-- Auxiliary lemma for `IsSFiniteKernel.withDensity`.
If a kernel `κ` is finite, then `withDensity κ f` is s-finite. -/
theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFiniteKernel κ]
    (hf_ne_top : ∀ a b, f a b ≠ ∞) : IsSFiniteKernel (withDensity κ f) := by
  -- We already have that for `f` bounded from above and a `κ` a finite kernel,
  -- `withDensity κ f` is finite. We write any function as a countable sum of bounded
  -- functions, and decompose an s-finite kernel as a sum of finite kernels. We then use that
  -- `withDensity` commutes with sums for both arguments and get a sum of finite kernels.
  by_cases hf : Measurable (Function.uncurry f)
  swap; · rw [withDensity_of_not_measurable _ hf]; infer_instance
  let fs : ℕ → α → β → ℝ≥0∞ := fun n a b => min (f a b) (n + 1) - min (f a b) n
  have h_le : ∀ a b n, ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ n := by
    intro a b n hn
    have : (f a b).toReal ≤ n := Nat.le_of_ceil_le hn
    rw [← ENNReal.le_ofReal_iff_toReal_le (hf_ne_top a b) _] at this
    · refine' this.trans (le_of_eq _)
      rw [ENNReal.ofReal_coe_nat]
    · norm_cast
      exact zero_le _
  have h_zero : ∀ a b n, ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0 := by
    intro a b n hn
    suffices min (f a b) (n + 1) = f a b ∧ min (f a b) n = f a b by
      simp_rw [this.1, this.2, tsub_self (f a b)]
    exact ⟨min_eq_left ((h_le a b n hn).trans (le_add_of_nonneg_right zero_le_one)),
      min_eq_left (h_le a b n hn)⟩
  have hf_eq_tsum : f = ∑' n, fs n := by
    have h_sum_a : ∀ a, Summable fun n => fs n a := by
      refine' fun a => Pi.summable.mpr fun b => _
      suffices : ∀ n, n ∉ Finset.range ⌈(f a b).toReal⌉₊ → fs n a b = 0
      exact summable_of_ne_finset_zero this
      intro n hn_not_mem
      rw [Finset.mem_range, not_lt] at hn_not_mem
      exact h_zero a b n hn_not_mem
    ext a b : 2
    rw [tsum_apply (Pi.summable.mpr h_sum_a), tsum_apply (h_sum_a a),
      ENNReal.tsum_eq_liminf_sum_nat]
    have h_finset_sum : ∀ n, ∑ i in Finset.range n, fs i a b = min (f a b) n := by
      intro n
      induction' n with n hn
      · simp
      rw [Finset.sum_range_succ, hn]
      simp
    simp_rw [h_finset_sum]
    refine' (Filter.Tendsto.liminf_eq _).symm
    refine' Filter.Tendsto.congr' _ tendsto_const_nhds
    rw [Filter.EventuallyEq, Filter.eventually_atTop]
    exact ⟨⌈(f a b).toReal⌉₊, fun n hn => (min_eq_left (h_le a b n hn)).symm⟩
  rw [hf_eq_tsum, withDensity_tsum _ fun n : ℕ => _]
  swap; · exact fun _ => (hf.min measurable_const).sub (hf.min measurable_const)
  refine' isSFiniteKernel_sum fun n => _
  

Proof state:
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f : α → β → ℝ≥0∞
κ : ↥(kernel α β)
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ⊤
hf : Measurable (Function.uncurry f)
fs : ℕ → α → β → ℝ≥0∞ := fun n a b => min (f a b) (↑n + 1) - min (f a b) ↑n
h_le : ∀ (a : α) (b : β) (n : ℕ), ⌈ENNReal.toReal (f a b)⌉₊ ≤ n → f a b ≤ ↑n
h_zero : ∀ (a : α) (b : β) (n : ℕ), ⌈ENNReal.toReal (f a b)⌉₊ ≤ n → fs n a b = 0
hf_eq_tsum : f = ∑' (n : ℕ), fs n
n : ℕ
⊢ IsSFiniteKernel (withDensity κ (fs n))

Next tactic:
suffices IsFiniteKernel (withDensity κ (fs n)) by haveI := this; infer_instance