Theorem: MeasureTheory.OuterMeasure.f_iUnion
File: Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean

Proof steps:
state_before: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), m.IsCaratheodory (s i)
hd : Pairwise (Disjoint on s)
⊢ m (⋃ i, s i) = ∑' (i : ℕ), m (s i)
state_after: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), m.IsCaratheodory (s i)
hd : Pairwise (Disjoint on s)
⊢ ∑' (i : ℕ), m (s i) ≤ m (⋃ i, s i)
tactic: refine le_antisymm (measure_iUnion_le s) ?_
state_before: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), m.IsCaratheodory (s i)
hd : Pairwise (Disjoint on s)
⊢ ∑' (i : ℕ), m (s i) ≤ m (⋃ i, s i)
state_after: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), m.IsCaratheodory (s i)
hd : Pairwise (Disjoint on s)
⊢ ⨆ i, ∑ a ∈ Finset.range i, m (s a) ≤ m (⋃ i, s i)
tactic: rw [ENNReal.tsum_eq_iSup_nat]
state_before: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), m.IsCaratheodory (s i)
hd : Pairwise (Disjoint on s)
⊢ ⨆ i, ∑ a ∈ Finset.range i, m (s a) ≤ m (⋃ i, s i)
state_after: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), m.IsCaratheodory (s i)
hd : Pairwise (Disjoint on s)
n : ℕ
⊢ ∑ a ∈ Finset.range n, m (s a) ≤ m (⋃ i, s i)
tactic: refine iSup_le fun n => ?_
state_before: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), m.IsCaratheodory (s i)
hd : Pairwise (Disjoint on s)
n : ℕ
⊢ ∑ a ∈ Finset.range n, m (s a) ≤ m (⋃ i, s i)
state_after: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), m.IsCaratheodory (s i)
hd : Pairwise (Disjoint on s)
n : ℕ
this : ∑ i ∈ Finset.range n, m (univ ∩ s i) = m (univ ∩ ⋃ i, ⋃ (_ : i < n), s i)
⊢ ∑ a ∈ Finset.range n, m (s a) ≤ m (⋃ i, s i)
tactic: have := @isCaratheodory_sum _ m _ h hd univ n
state_before: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), m.IsCaratheodory (s i)
hd : Pairwise (Disjoint on s)
n : ℕ
this : ∑ i ∈ Finset.range n, m (univ ∩ s i) = m (univ ∩ ⋃ i, ⋃ (_ : i < n), s i)
⊢ ∑ a ∈ Finset.range n, m (s a) ≤ m (⋃ i, s i)
state_after: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), m.IsCaratheodory (s i)
hd : Pairwise (Disjoint on s)
n : ℕ
this : ∑ x ∈ Finset.range n, m (s x) = m (⋃ i, ⋃ (_ : i < n), s i)
⊢ ∑ a ∈ Finset.range n, m (s a) ≤ m (⋃ i, s i)
tactic: simp only [inter_comm, inter_univ, univ_inter] at this
state_before: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), m.IsCaratheodory (s i)
hd : Pairwise (Disjoint on s)
n : ℕ
this : ∑ x ∈ Finset.range n, m (s x) = m (⋃ i, ⋃ (_ : i < n), s i)
⊢ ∑ a ∈ Finset.range n, m (s a) ≤ m (⋃ i, s i)
state_after: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), m.IsCaratheodory (s i)
hd : Pairwise (Disjoint on s)
n : ℕ
this : ∑ x ∈ Finset.range n, m (s x) = m (⋃ i, ⋃ (_ : i < n), s i)
⊢ m (⋃ i, ⋃ (_ : i < n), s i) ≤ m (⋃ i, s i)
tactic: simp only [this]
state_before: α : Type u
m : OuterMeasure α
s✝ s₁ s₂ : Set α
s : ℕ → Set α
h : ∀ (i : ℕ), m.IsCaratheodory (s i)
hd : Pairwise (Disjoint on s)
n : ℕ
this : ∑ x ∈ Finset.range n, m (s x) = m (⋃ i, ⋃ (_ : i < n), s i)
⊢ m (⋃ i, ⋃ (_ : i < n), s i) ≤ m (⋃ i, s i)
state_after: no goals
tactic: exact m.mono (iUnion₂_subset fun i _ => subset_iUnion _ i)