-- 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:
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 < ⊤

Tactic:
refine (?_ : ⨆ (s) (_ : MeasurableSet s)
  (_ : SigmaFinite (μ.restrict s)), ν s ≤ ν Set.univ).trans_lt (measure_lt_top _ _)

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
⊢ ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s ≤ ν Set.univ

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
s : Set α
⊢ ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s ≤ ν Set.univ

Tactic:
exact iSup_le (fun _ ↦ iSup_le (fun _ ↦ measure_mono (Set.subset_univ s)))

Result:
no goals

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
ht_meas : MeasurableSet t
⊢ SigmaFinite (μ.restrict 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
ht_meas : MeasurableSet t
h_not_mem : ¬SigmaFinite (μ.restrict t)
⊢ False

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
⊢ 0 ≤ ν ∅

Tactic:
exact zero_le'

Result:
no goals

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
⊢ SigmaFinite 0

Tactic:
infer_instance

Result:
no goals

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
⊢ ∃ t,
    MeasurableSet t ∧
      SigmaFinite (μ.restrict t) ∧
        (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n ≤ ν t

Tactic:
have ht_meas : MeasurableSet t := by
  by_contra h_not_mem
  simp only [h_not_mem] at ht
  simp 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 :
  (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n <
    ⨆ (_ : MeasurableSet t), ⨆ (_ : SigmaFinite (μ.restrict t)), ν t
ht_meas : MeasurableSet 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
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 < ⊤
⊢ 1 / ↑n ≠ 0

Tactic:
simp

Result:
no goals