case intro
α : Type u_2
β : Type ?u.14897
γ : Type ?u.14900
δ : Type ?u.14903
ι✝ : Type ?u.14906
R : Type ?u.14909
R' : Type ?u.14912
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s s₁ s₂ t : Set α
ι : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), MeasurableSet (As i)
As_disj : Pairwise (Disjoint on As)
S : ℝ≥0∞
hS : HasSum (fun i => ↑↑μ (As i)) S
⊢ (∑' (i : ι), ↑↑μ (As i)) ≤ ↑↑μ (⋃ (i : ι), As i)
α : Type u_2
β : Type ?u.14897
γ : Type ?u.14900
δ : Type ?u.14903
ι✝ : Type ?u.14906
R : Type ?u.14909
R' : Type ?u.14912
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s s₁ s₂ t : Set α
ι : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), MeasurableSet (As i)
As_disj : Pairwise (Disjoint on As)
⊢ (∑' (i : ι), ↑↑μ (As i)) ≤ ↑↑μ (⋃ (i : ι), As i)
rcases show Summable fun i => μ (As i) from ENNReal.summable with ⟨S, hS⟩
case intro
α : Type u_2
β : Type ?u.14897
γ : Type ?u.14900
δ : Type ?u.14903
ι✝ : Type ?u.14906
R : Type ?u.14909
R' : Type ?u.14912
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s s₁ s₂ t : Set α
ι : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), MeasurableSet (As i)
As_disj : Pairwise (Disjoint on As)
S : ℝ≥0∞
hS : HasSum (fun i => ↑↑μ (As i)) S
⊢ S ≤ ↑↑μ (⋃ (i : ι), As i)
case intro
α : Type u_2
β : Type ?u.14897
γ : Type ?u.14900
δ : Type ?u.14903
ι✝ : Type ?u.14906
R : Type ?u.14909
R' : Type ?u.14912
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s s₁ s₂ t : Set α
ι : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), MeasurableSet (As i)
As_disj : Pairwise (Disjoint on As)
S : ℝ≥0∞
hS : HasSum (fun i => ↑↑μ (As i)) S
⊢ (∑' (i : ι), ↑↑μ (As i)) ≤ ↑↑μ (⋃ (i : ι), As i)
rw [hS.tsum_eq]
case intro
α : Type u_2
β : Type ?u.14897
γ : Type ?u.14900
δ : Type ?u.14903
ι✝ : Type ?u.14906
R : Type ?u.14909
R' : Type ?u.14912
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s s₁ s₂ t : Set α
ι : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), MeasurableSet (As i)
As_disj : Pairwise (Disjoint on As)
S : ℝ≥0∞
hS : HasSum (fun i => ↑↑μ (As i)) S
⊢ ∀ (x : Finset ι), (fun s => ∑ b in s, (fun i => ↑↑μ (As i)) b) x ≤ (fun x => ↑↑μ (⋃ (i : ι), As i)) x
case intro
α : Type u_2
β : Type ?u.14897
γ : Type ?u.14900
δ : Type ?u.14903
ι✝ : Type ?u.14906
R : Type ?u.14909
R' : Type ?u.14912
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s s₁ s₂ t : Set α
ι : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), MeasurableSet (As i)
As_disj : Pairwise (Disjoint on As)
S : ℝ≥0∞
hS : HasSum (fun i => ↑↑μ (As i)) S
⊢ S ≤ ↑↑μ (⋃ (i : ι), As i)
refine' tendsto_le_of_eventuallyLE hS tendsto_const_nhds (eventually_of_forall _)
case intro
α : Type u_2
β : Type ?u.14897
γ : Type ?u.14900
δ : Type ?u.14903
ι✝ : Type ?u.14906
R : Type ?u.14909
R' : Type ?u.14912
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), MeasurableSet (As i)
As_disj : Pairwise (Disjoint on As)
S : ℝ≥0∞
hS : HasSum (fun i => ↑↑μ (As i)) S
s : Finset ι
⊢ (fun s => ∑ b in s, (fun i => ↑↑μ (As i)) b) s ≤ (fun x => ↑↑μ (⋃ (i : ι), As i)) s
case intro
α : Type u_2
β : Type ?u.14897
γ : Type ?u.14900
δ : Type ?u.14903
ι✝ : Type ?u.14906
R : Type ?u.14909
R' : Type ?u.14912
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s s₁ s₂ t : Set α
ι : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), MeasurableSet (As i)
As_disj : Pairwise (Disjoint on As)
S : ℝ≥0∞
hS : HasSum (fun i => ↑↑μ (As i)) S
⊢ ∀ (x : Finset ι), (fun s => ∑ b in s, (fun i => ↑↑μ (As i)) b) x ≤ (fun x => ↑↑μ (⋃ (i : ι), As i)) x
intro s
case intro
α : Type u_2
β : Type ?u.14897
γ : Type ?u.14900
δ : Type ?u.14903
ι✝ : Type ?u.14906
R : Type ?u.14909
R' : Type ?u.14912
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), MeasurableSet (As i)
As_disj : Pairwise (Disjoint on As)
S : ℝ≥0∞
hS : HasSum (fun i => ↑↑μ (As i)) S
s : Finset ι
⊢ ↑↑μ (⋃ (b : ι) (_ : b ∈ s), As b) ≤ ↑↑μ (⋃ (i : ι), As i)
case intro
α : Type u_2
β : Type ?u.14897
γ : Type ?u.14900
δ : Type ?u.14903
ι✝ : Type ?u.14906
R : Type ?u.14909
R' : Type ?u.14912
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), MeasurableSet (As i)
As_disj : Pairwise (Disjoint on As)
S : ℝ≥0∞
hS : HasSum (fun i => ↑↑μ (As i)) S
s : Finset ι
⊢ (fun s => ∑ b in s, (fun i => ↑↑μ (As i)) b) s ≤ (fun x => ↑↑μ (⋃ (i : ι), As i)) s
simp [← measure_biUnion_finset (fun _i _hi _j _hj hij => As_disj hij) fun i _ => As_mble i]
no goals
case intro
α : Type u_2
β : Type ?u.14897
γ : Type ?u.14900
δ : Type ?u.14903
ι✝ : Type ?u.14906
R : Type ?u.14909
R' : Type ?u.14912
m : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), MeasurableSet (As i)
As_disj : Pairwise (Disjoint on As)
S : ℝ≥0∞
hS : HasSum (fun i => ↑↑μ (As i)) S
s : Finset ι
⊢ ↑↑μ (⋃ (b : ι) (_ : b ∈ s), As b) ≤ ↑↑μ (⋃ (i : ι), As i)
exact measure_mono (iUnion₂_subset_iUnion (fun i : ι => i ∈ s) fun i : ι => As i)