/-- If a continuous function `c` realizes its maximum at a unique point `x₀` in a compact set `s`,
then the sequence of functions `(c x) ^ n / ∫ (c x) ^ n` is a sequence of peak functions
concentrating around `x₀`. Therefore, `∫ (c x) ^ n * g / ∫ (c x) ^ n` converges to `g x₀` if `g` is
integrable on `s` and continuous at `x₀`.

Version assuming that `μ` gives positive mass to all neighborhoods of `x₀` within `s`.
For a less precise but more usable version, see
`tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_continuousOn`.
 -/
theorem tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_measure_nhdsWithin_pos
    [MetrizableSpace α] [IsLocallyFiniteMeasure μ] (hs : IsCompact s)
    (hμ : ∀ u, IsOpen u → x₀ ∈ u → 0 < μ (u ∩ s)) {c : α → ℝ} (hc : ContinuousOn c s)
    (h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀) (hnc : ∀ x ∈ s, 0 ≤ c x) (hnc₀ : 0 < c x₀) (h₀ : x₀ ∈ s)
    (hmg : IntegrableOn g s μ) (hcg : ContinuousWithinAt g s x₀) :
    Tendsto (fun n : ℕ => (∫ x in s, c x ^ n ∂μ)⁻¹ • ∫ x in s, c x ^ n • g x ∂μ)
      atTop (𝓝 (g x₀)) := by
  /- We apply the general result
    `tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt` to the sequence of
    peak functions `φₙ = (c x) ^ n / ∫ (c x) ^ n`. The only nontrivial bit is to check that this
    sequence converges uniformly to zero on any set `s \ u` away from `x₀`. By compactness, the
    function `c` is bounded by `t < c x₀` there. Consider `t' ∈ (t, c x₀)`, and a neighborhood `v`
    of `x₀` where `c x ≥ t'`, by continuity. Then `∫ (c x) ^ n` is bounded below by `t' ^ n μ v`.
    It follows that, on `s \ u`, then `φₙ x ≤ t ^ n / (t' ^ n μ v)`,
    which tends (exponentially fast) to zero with `n`. -/
  let φ : ℕ → α → ℝ := fun n x => (∫ x in s, c x ^ n ∂μ)⁻¹ * c x ^ n
  have hnφ : ∀ n, ∀ x ∈ s, 0 ≤ φ n x := by
    intro n x hx
    apply mul_nonneg (inv_nonneg.2 _) (pow_nonneg (hnc x hx) _)
    exact set_integral_nonneg hs.measurableSet fun x hx => pow_nonneg (hnc x hx) _
  have I : ∀ n, IntegrableOn (fun x => c x ^ n) s μ := fun n =>
    ContinuousOn.integrableOn_compact hs (hc.pow n)
  have J : ∀ n, 0 ≤ᵐ[μ.restrict s] fun x : α => c x ^ n := by
    intro n
    filter_upwards [ae_restrict_mem hs.measurableSet] with x hx
    exact pow_nonneg (hnc x hx) n
  have P : ∀ n, (0 : ℝ) < ∫ x in s, c x ^ n ∂μ := by
    intro n
    refine' (set_integral_pos_iff_support_of_nonneg_ae (J n) (I n)).2 _
    obtain ⟨u, u_open, x₀_u, hu⟩ : ∃ u : Set α, IsOpen u ∧ x₀ ∈ u ∧ u ∩ s ⊆ c ⁻¹' Ioi 0 :=
      _root_.continuousOn_iff.1 hc x₀ h₀ (Ioi (0 : ℝ)) isOpen_Ioi hnc₀
    apply (hμ u u_open x₀_u).trans_le
    exact measure_mono fun x hx => ⟨ne_of_gt (pow_pos (a := c x) (hu hx) _), hx.2⟩
  have hiφ : ∀ n, ∫ x in s, φ n x ∂μ = 1 := fun n => by
    rw [integral_mul_left, inv_mul_cancel (P n).ne']
  have A : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 atTop (s \ u) := by
    intro u u_open x₀u
    obtain ⟨t, t_pos, tx₀, ht⟩ : ∃ t, 0 ≤ t ∧ t < c x₀ ∧ ∀ x ∈ s \ u, c x ≤ t := by
      rcases eq_empty_or_nonempty (s \ u) with (h | h)
      · exact
          ⟨0, le_rfl, hnc₀, by simp only [h, mem_empty_iff_false, IsEmpty.forall_iff, imp_true_iff]⟩
      obtain ⟨x, hx, h'x⟩ : ∃ x ∈ s \ u, ∀ y ∈ s \ u, c y ≤ c x :=
        IsCompact.exists_isMaxOn (hs.diff u_open) h (hc.mono (diff_subset _ _))
      refine' ⟨c x, hnc x hx.1, h'c x hx.1 _, h'x⟩
      rintro rfl
      exact hx.2 x₀u
    obtain ⟨t', tt', t'x₀⟩ : ∃ t', t < t' ∧ t' < c x₀ := exists_between tx₀
    have t'_pos : 0 < t' := t_pos.trans_lt tt'
    obtain ⟨v, v_open, x₀_v, hv⟩ : ∃ v : Set α, IsOpen v ∧ x₀ ∈ v ∧ v ∩ s ⊆ c ⁻¹' Ioi t' :=
      _root_.continuousOn_iff.1 hc x₀ h₀ (Ioi t') isOpen_Ioi t'x₀
    have M : ∀ n, ∀ x ∈ s \ u, φ n x ≤ (μ (v ∩ s)).toReal⁻¹ * (t / t') ^ n := by
      intro n x hx
      have B : t' ^ n * (μ (v ∩ s)).toReal ≤ ∫ y in s, c y ^ n ∂μ :=
        calc
          t' ^ n * (μ (v ∩ s)).toReal = ∫ _ in v ∩ s, t' ^ n ∂μ := by
            simp only [integral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter,
              Algebra.id.smul_eq_mul, mul_comm]
          _ ≤ ∫ y in v ∩ s, c y ^ n ∂μ := by
            apply set_integral_mono_on _ _ (v_open.measurableSet.inter hs.measurableSet) _
            · apply integrableOn_const.2 (Or.inr _)
              exact lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) hs.measure_lt_top
            · exact (I n).mono (inter_subset_right _ _) le_rfl
            · intro x hx
              exact pow_le_pow_left t'_pos.le (le_of_lt (hv hx)) _
          _ ≤ ∫ y in s, c y ^ n ∂μ :=
            set_integral_mono_set (I n) (J n) (eventually_of_forall (inter_subset_right _ _))
      simp_rw [← div_eq_inv_mul, div_pow, div_div]
      apply div_le_div (pow_nonneg t_pos n) _ _ B
      · exact pow_le_pow_left (hnc _ hx.1) (ht x hx) _
      · apply mul_pos (pow_pos (t_pos.trans_lt tt') _) (ENNReal.toReal_pos (hμ v v_open x₀_v).ne' _)
        have : μ (v ∩ s) ≤ μ s := measure_mono (inter_subset_right _ _)
        exact ne_of_lt (lt_of_le_of_lt this hs.measure_lt_top)
    have N :
      Tendsto (fun n => (μ (v ∩ s)).toReal⁻¹ * (t / t') ^ n) atTop
        (𝓝 ((μ (v ∩ s)).toReal⁻¹ * 0)) := by
      apply Tendsto.mul tendsto_const_nhds _
      apply tendsto_pow_atTop_nhds_0_of_lt_1 (div_nonneg t_pos t'_pos.le)
      exact (div_lt_one t'_pos).2 tt'
    rw [mul_zero] at N
    refine' tendstoUniformlyOn_iff.2 fun ε εpos => _
    filter_upwards [(tendsto_order.1 N).2 ε εpos] with n hn x hx
    simp only [Pi.zero_apply, dist_zero_left, Real.norm_of_nonneg (hnφ n x hx.1)]
    

Proof state:
case h
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : BorelSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s : Set α
φ✝ : ι → α → ℝ
inst✝² : CompleteSpace E
inst✝¹ : MetrizableSpace α
inst✝ : IsLocallyFiniteMeasure μ
hs : IsCompact s
hμ : ∀ (u : Set α), IsOpen u → x₀ ∈ u → 0 < ↑↑μ (u ∩ s)
c : α → ℝ
hc : ContinuousOn c s
h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀
hnc : ∀ x ∈ s, 0 ≤ c x
hnc₀ : 0 < c x₀
h₀ : x₀ ∈ s
hmg : IntegrableOn g s
hcg : ContinuousWithinAt g s x₀
φ : ℕ → α → ℝ := fun n x => (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ * c x ^ n
hnφ : ∀ (n : ℕ), ∀ x ∈ s, 0 ≤ φ n x
I : ∀ (n : ℕ), IntegrableOn (fun x => c x ^ n) s
J : ∀ (n : ℕ), 0 ≤ᶠ[ae (Measure.restrict μ s)] fun x => c x ^ n
P : ∀ (n : ℕ), 0 < ∫ (x : α) in s, c x ^ n ∂μ
hiφ : ∀ (n : ℕ), ∫ (x : α) in s, φ n x ∂μ = 1
u : Set α
u_open : IsOpen u
x₀u : x₀ ∈ u
t : ℝ
t_pos : 0 ≤ t
tx₀ : t < c x₀
ht : ∀ x ∈ s \ u, c x ≤ t
t' : ℝ
tt' : t < t'
t'x₀ : t' < c x₀
t'_pos : 0 < t'
v : Set α
v_open : IsOpen v
x₀_v : x₀ ∈ v
hv : v ∩ s ⊆ c ⁻¹' Ioi t'
M : ∀ (n : ℕ), ∀ x ∈ s \ u, φ n x ≤ (ENNReal.toReal (↑↑μ (v ∩ s)))⁻¹ * (t / t') ^ n
N : Tendsto (fun n => (ENNReal.toReal (↑↑μ (v ∩ s)))⁻¹ * (t / t') ^ n) atTop (𝓝 0)
ε : ℝ
εpos : ε > 0
n : ℕ
hn : (ENNReal.toReal (↑↑μ (v ∩ s)))⁻¹ * (t / t') ^ n < ε
x : α
hx : x ∈ s \ u
⊢ (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ * c x ^ n < ε

Next tactic:
exact (M n x hx).trans_lt hn