Theorem: MeasureTheory.tsum_meas_le_meas_iUnion_of_disjoint₀
File: Mathlib/MeasureTheory/Measure/MeasureSpace.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s s₁ s₂ t : Set α
ι : Type u_8
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
⊢ ∑' (i : ι), μ (As i) ≤ μ (⋃ i, As i)
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s s₁ s₂ t : Set α
ι : Type u_8
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
⊢ ∀ (i : Finset ι), ∑ a ∈ i, μ (As a) ≤ μ (⋃ i, As i)
tactic: rw [ENNReal.tsum_eq_iSup_sum, iSup_le_iff]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s s₁ s₂ t : Set α
ι : Type u_8
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
⊢ ∀ (i : Finset ι), ∑ a ∈ i, μ (As a) ≤ μ (⋃ i, As i)
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_8
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
s : Finset ι
⊢ ∑ a ∈ s, μ (As a) ≤ μ (⋃ i, As i)
tactic: intro s
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_8
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
s : Finset ι
⊢ ∑ a ∈ s, μ (As a) ≤ μ (⋃ i, As i)
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_8
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
s : Finset ι
⊢ μ (⋃ b ∈ s, As b) ≤ μ (⋃ i, As i)
tactic: simp only [← measure_biUnion_finset₀ (fun _i _hi _j _hj hij => As_disj hij) fun i _ => As_mble i]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_8
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
s : Finset ι
⊢ μ (⋃ b ∈ s, As b) ≤ μ (⋃ i, As i)
state_after: case h.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_8
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
s : Finset ι
i✝ : ι
⊢ ⋃ (_ : i✝ ∈ s), As i✝ ⊆ As i✝
tactic: gcongr
state_before: case h.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_8
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
s : Finset ι
i✝ : ι
⊢ ⋃ (_ : i✝ ∈ s), As i✝ ⊆ As i✝
state_after: no goals
tactic: exact iUnion_subset fun _ ↦ Subset.rfl