-- Mathlib/MeasureTheory/Function/Intersectivity.lean
/-- **Bergelson Intersectivity Lemma**: In a finite measure space, a sequence of events that have
measure at least `r` has an infinite subset whose finite intersections all have positive volume.

TODO: The infinity of `t` should be strengthened to `t` having positive natural density. -/
lemma bergelson' {s : ℕ → Set α} (hs : ∀ n, MeasurableSet (s n)) (hr₀ : r ≠ 0)
    (hr : ∀ n, r ≤ μ (s n)) :
    ∃ t : Set ℕ, t.Infinite ∧ ∀ ⦃u⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)

Proof state:
case intro.intro.refine_2.intro
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun x_1 => f x_1 x) atTop ∂μ ≤ μ univ
x : α
hxN : x ∉ N
hx : ⨍⁻ (a : α), limsup (fun x => f x a) atTop ∂μ ≤ limsup (fun x_1 => f x_1 x) atTop
b : ℝ≥0∞
hb : ∀ᶠ (n : ℕ) in atTop, ⨍⁻ (x : α), f n x ∂μ ≤ b
n : ℕ
hn : ⨍⁻ (x : α), f n x ∂μ ≤ b
⊢ r / μ univ ≤ b

Tactic:
rw [laverage_eq] at hn

Result:
case intro.intro.refine_2.intro
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun x_1 => f x_1 x) atTop ∂μ ≤ μ univ
x : α
hxN : x ∉ N
hx : ⨍⁻ (a : α), limsup (fun x => f x a) atTop ∂μ ≤ limsup (fun x_1 => f x_1 x) atTop
b : ℝ≥0∞
hb : ∀ᶠ (n : ℕ) in atTop, ⨍⁻ (x : α), f n x ∂μ ≤ b
n : ℕ
hn : (∫⁻ (x : α), f n x ∂μ) / μ univ ≤ b
⊢ r / μ univ ≤ b

Proof state:
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)

Tactic:
let M (f : α → ℝ) : Set α := {x | eLpNormEssSup f μ < ‖f x‖₊}

Result:
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}
⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)

Proof state:
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
n : ℕ
⊢ ↑(n + 1) ≠ 0 ∨ ∑ x ∈ Finset.range (n + 1), μ (s x) ≠ 0

Tactic:
simp

Result:
no goals

Proof state:
case intro.intro.refine_3.refine_1.refine_2
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun x_1 => f x_1 x) atTop ∂μ ≤ μ univ
x : α
hxN : x ∉ N
hx : r / μ univ ≤ limsup (fun x_1 => f x_1 x) atTop
hxs : {n | x ∈ s n}.Finite
n : ℕ
⊢ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1) x ≤ ↑hxs.toFinset.card

Tactic:
classical
simpa only [Finset.sum_apply, indicator_apply, Pi.one_apply, Finset.sum_boole, Nat.cast_le]
  using Finset.card_le_card fun m hm ↦ hxs.mem_toFinset.2 (Finset.mem_filter.1 hm).2

Result:
no goals

Proof state:
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
hf : ∀ (n : ℕ), Measurable (f n)
⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)

Tactic:
have hf₁ n : f n ≤ 1 := by
  rintro a
  rw [hfapp, ← ENNReal.div_eq_inv_mul]
  refine (ENNReal.div_le_iff_le_mul (Or.inl <| Nat.cast_ne_zero.2 n.succ_ne_zero) <|
    Or.inr one_ne_zero).2 ?_
  rw [mul_comm, ← nsmul_eq_mul, ← Finset.card_range n.succ]
  exact Finset.sum_le_card_nsmul _ _ _ fun _ _ ↦ indicator_le (fun _ _ ↦ le_rfl) _

Result:
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)

Proof state:
case intro.intro.refine_3.refine_2
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun x_1 => f x_1 x) atTop ∂μ ≤ μ univ
x : α
hxN : x ∉ N
hx : r / μ univ ≤ limsup (fun x_1 => f x_1 x) atTop
u : Set ℕ
hux : u ⊆ {n | x ∈ s n}
hu : u.Finite
⊢ 0 < μ (⋂ n ∈ hu.toFinset, s n)

Tactic:
exact hN₁ _ ⟨x, mem_iInter₂.2 fun n hn ↦ hux <| hu.mem_toFinset.1 hn, hxN⟩

Result:
no goals