-- Mathlib/MeasureTheory/Decomposition/Exhaustion.lean
/-- Let `C` be the supremum of `ν s` over all measurable sets `s` such that `μ.restrict s` is
sigma-finite. `C` is finite since `ν` is a finite measure. Then there exists a measurable set `t`
with `μ.restrict t` sigma-finite such that `ν t ≥ C - 1/n`. -/
lemma exists_isSigmaFiniteSet_measure_ge (μ ν : Measure α) [IsFiniteMeasure ν] (n : ℕ) :
    ∃ t, MeasurableSet t ∧ SigmaFinite (μ.restrict t)
      ∧ (⨆ (s) (_ : MeasurableSet s) (_ : SigmaFinite (μ.restrict s)), ν s) - 1/n ≤ ν t

Proof state:
case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
hC_lt : ¬1 / ↑n < ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s
⊢ (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n ≤ ν ∅

Tactic:
rw [tsub_eq_zero_of_le (not_lt.mp hC_lt)]

Result:
case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
hC_lt : ¬1 / ↑n < ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s
⊢ 0 ≤ ν ∅

Proof state:
case pos.intro
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
hC_lt : 1 / ↑n < ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s
h_lt_top : ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s < ⊤
t : Set α
ht :
  (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n <
    ⨆ (_ : MeasurableSet t), ⨆ (_ : SigmaFinite (μ.restrict t)), ν t
ht_meas : MeasurableSet t
ht_mem : SigmaFinite (μ.restrict t)
⊢ (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n ≤ ν t

Tactic:
simp only [ht_meas, ht_mem, iSup_true] at ht

Result:
case pos.intro
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
hC_lt : 1 / ↑n < ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s
h_lt_top : ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s < ⊤
t : Set α
ht_meas : MeasurableSet t
ht_mem : SigmaFinite (μ.restrict t)
ht : (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n < ν t
⊢ (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n ≤ ν t

Proof state:
case pos.intro
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
hC_lt : 1 / ↑n < ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s
h_lt_top : ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s < ⊤
t : Set α
ht :
  (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n <
    ⨆ (_ : MeasurableSet t), ⨆ (_ : SigmaFinite (μ.restrict t)), ν t
ht_meas : MeasurableSet t
ht_mem : SigmaFinite (μ.restrict t)
⊢ ∃ t,
    MeasurableSet t ∧
      SigmaFinite (μ.restrict t) ∧
        (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n ≤ ν t

Tactic:
refine ⟨t, ht_meas, ht_mem, ?_⟩

Result:
case pos.intro
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
hC_lt : 1 / ↑n < ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s
h_lt_top : ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s < ⊤
t : Set α
ht :
  (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n <
    ⨆ (_ : MeasurableSet t), ⨆ (_ : SigmaFinite (μ.restrict t)), ν t
ht_meas : MeasurableSet t
ht_mem : SigmaFinite (μ.restrict t)
⊢ (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n ≤ ν t

Proof state:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
hC_lt : 1 / ↑n < ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s
h_lt_top : ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s < ⊤
t : Set α
ht :
  (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n <
    ⨆ (_ : MeasurableSet t), ⨆ (_ : SigmaFinite (μ.restrict t)), ν t
⊢ MeasurableSet t

Tactic:
by_contra h_not_mem

Result:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
hC_lt : 1 / ↑n < ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s
h_lt_top : ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s < ⊤
t : Set α
ht :
  (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n <
    ⨆ (_ : MeasurableSet t), ⨆ (_ : SigmaFinite (μ.restrict t)), ν t
h_not_mem : ¬MeasurableSet t
⊢ False

Proof state:
case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
hC_lt : 1 / ↑n < ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s
h_lt_top : ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s < ⊤
⊢ ∃ t,
    MeasurableSet t ∧
      SigmaFinite (μ.restrict t) ∧
        (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n ≤ ν t

Tactic:
obtain ⟨t, ht⟩ := exists_lt_of_lt_ciSup
  (ENNReal.sub_lt_self h_lt_top.ne (ne_zero_of_lt hC_lt) (by simp) :
      (⨆ (s) (_ : MeasurableSet s) (_ : SigmaFinite (μ.restrict s)), ν s) - 1/n
    < ⨆ (s) (_ : MeasurableSet s) (_ : SigmaFinite (μ.restrict s)), ν s)

Result:
case pos.intro
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
hC_lt : 1 / ↑n < ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s
h_lt_top : ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s < ⊤
t : Set α
ht :
  (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n <
    ⨆ (_ : MeasurableSet t), ⨆ (_ : SigmaFinite (μ.restrict t)), ν t
⊢ ∃ t,
    MeasurableSet t ∧
      SigmaFinite (μ.restrict t) ∧
        (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n ≤ ν t

Proof state:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
⊢ ∃ t,
    MeasurableSet t ∧
      SigmaFinite (μ.restrict t) ∧
        (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n ≤ ν t

Tactic:
by_cases hC_lt : 1/n < ⨆ (s) (_ : MeasurableSet s) (_ : SigmaFinite (μ.restrict s)), ν s

Result:
case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
hC_lt : 1 / ↑n < ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s
⊢ ∃ t,
    MeasurableSet t ∧
      SigmaFinite (μ.restrict t) ∧
        (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n ≤ ν t

case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
hC_lt : ¬1 / ↑n < ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s
⊢ ∃ t,
    MeasurableSet t ∧
      SigmaFinite (μ.restrict t) ∧
        (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n ≤ ν t

Proof state:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
hC_lt : 1 / ↑n < ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s
h_lt_top : ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s < ⊤
t : Set α
ht :
  (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n <
    ⨆ (_ : MeasurableSet t), ⨆ (_ : SigmaFinite (μ.restrict t)), ν t
h_not_mem : ¬MeasurableSet t
⊢ False

Tactic:
simp only [h_not_mem] at ht

Result:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
hC_lt : 1 / ↑n < ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s
h_lt_top : ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s < ⊤
t : Set α
h_not_mem : ¬MeasurableSet t
ht :
  (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n <
    ⨆ (_ : False), ⨆ (_ : SigmaFinite (μ.restrict t)), ν t
⊢ False