α : Type u_1
β : Type ?u.27550
γ : Type ?u.27553
δ : Type ?u.27556
ι : Type ?u.27559
R : Type ?u.27562
R' : Type ?u.27565
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s s₁✝ s₂✝ t s₁ s₂ s₃ : Set α
h12 : s₁ ⊆ s₂
h23 : s₂ ⊆ s₃
h_nulldiff : ↑↑μ (s₃ \ s₁) = 0
le12 : ↑↑μ s₁ ≤ ↑↑μ s₂
⊢ ↑↑μ s₁ = ↑↑μ s₂ ∧ ↑↑μ s₂ = ↑↑μ s₃
α : Type u_1
β : Type ?u.27550
γ : Type ?u.27553
δ : Type ?u.27556
ι : Type ?u.27559
R : Type ?u.27562
R' : Type ?u.27565
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s s₁✝ s₂✝ t s₁ s₂ s₃ : Set α
h12 : s₁ ⊆ s₂
h23 : s₂ ⊆ s₃
h_nulldiff : ↑↑μ (s₃ \ s₁) = 0
⊢ ↑↑μ s₁ = ↑↑μ s₂ ∧ ↑↑μ s₂ = ↑↑μ s₃
have le12 : μ s₁ ≤ μ s₂ := measure_mono h12
α : Type u_1
β : Type ?u.27550
γ : Type ?u.27553
δ : Type ?u.27556
ι : Type ?u.27559
R : Type ?u.27562
R' : Type ?u.27565
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s s₁✝ s₂✝ t s₁ s₂ s₃ : Set α
h12 : s₁ ⊆ s₂
h23 : s₂ ⊆ s₃
h_nulldiff : ↑↑μ (s₃ \ s₁) = 0
le12 : ↑↑μ s₁ ≤ ↑↑μ s₂
le23 : ↑↑μ s₂ ≤ ↑↑μ s₃
⊢ ↑↑μ s₁ = ↑↑μ s₂ ∧ ↑↑μ s₂ = ↑↑μ s₃
α : Type u_1
β : Type ?u.27550
γ : Type ?u.27553
δ : Type ?u.27556
ι : Type ?u.27559
R : Type ?u.27562
R' : Type ?u.27565
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s s₁✝ s₂✝ t s₁ s₂ s₃ : Set α
h12 : s₁ ⊆ s₂
h23 : s₂ ⊆ s₃
h_nulldiff : ↑↑μ (s₃ \ s₁) = 0
le12 : ↑↑μ s₁ ≤ ↑↑μ s₂
⊢ ↑↑μ s₁ = ↑↑μ s₂ ∧ ↑↑μ s₂ = ↑↑μ s₃
have le23 : μ s₂ ≤ μ s₃ := measure_mono h23
α : Type u_1
β : Type ?u.27550
γ : Type ?u.27553
δ : Type ?u.27556
ι : Type ?u.27559
R : Type ?u.27562
R' : Type ?u.27565
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s s₁✝ s₂✝ t s₁ s₂ s₃ : Set α
h12 : s₁ ⊆ s₂
h23 : s₂ ⊆ s₃
h_nulldiff : ↑↑μ (s₃ \ s₁) = 0
le12 : ↑↑μ s₁ ≤ ↑↑μ s₂
le23 : ↑↑μ s₂ ≤ ↑↑μ s₃
key : ↑↑μ s₃ ≤ ↑↑μ s₁
⊢ ↑↑μ s₁ = ↑↑μ s₂ ∧ ↑↑μ s₂ = ↑↑μ s₃
α : Type u_1
β : Type ?u.27550
γ : Type ?u.27553
δ : Type ?u.27556
ι : Type ?u.27559
R : Type ?u.27562
R' : Type ?u.27565
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s s₁✝ s₂✝ t s₁ s₂ s₃ : Set α
h12 : s₁ ⊆ s₂
h23 : s₂ ⊆ s₃
h_nulldiff : ↑↑μ (s₃ \ s₁) = 0
le12 : ↑↑μ s₁ ≤ ↑↑μ s₂
le23 : ↑↑μ s₂ ≤ ↑↑μ s₃
⊢ ↑↑μ s₁ = ↑↑μ s₂ ∧ ↑↑μ s₂ = ↑↑μ s₃
have key : μ s₃ ≤ μ s₁ :=
  calc
    μ s₃ = μ (s₃ \ s₁ ∪ s₁) := by rw [diff_union_of_subset (h12.trans h23)]
    _ ≤ μ (s₃ \ s₁) + μ s₁ := (measure_union_le _ _)
    _ = μ s₁ := by simp only [h_nulldiff, zero_add]
no goals
α : Type u_1
β : Type ?u.27550
γ : Type ?u.27553
δ : Type ?u.27556
ι : Type ?u.27559
R : Type ?u.27562
R' : Type ?u.27565
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s s₁✝ s₂✝ t s₁ s₂ s₃ : Set α
h12 : s₁ ⊆ s₂
h23 : s₂ ⊆ s₃
h_nulldiff : ↑↑μ (s₃ \ s₁) = 0
le12 : ↑↑μ s₁ ≤ ↑↑μ s₂
le23 : ↑↑μ s₂ ≤ ↑↑μ s₃
key : ↑↑μ s₃ ≤ ↑↑μ s₁
⊢ ↑↑μ s₁ = ↑↑μ s₂ ∧ ↑↑μ s₂ = ↑↑μ s₃
exact ⟨le12.antisymm (le23.trans key), le23.antisymm (key.trans le12)⟩
no goals
α : Type u_1
β : Type ?u.27550
γ : Type ?u.27553
δ : Type ?u.27556
ι : Type ?u.27559
R : Type ?u.27562
R' : Type ?u.27565
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s s₁✝ s₂✝ t s₁ s₂ s₃ : Set α
h12 : s₁ ⊆ s₂
h23 : s₂ ⊆ s₃
h_nulldiff : ↑↑μ (s₃ \ s₁) = 0
le12 : ↑↑μ s₁ ≤ ↑↑μ s₂
le23 : ↑↑μ s₂ ≤ ↑↑μ s₃
⊢ ↑↑μ s₃ = ↑↑μ (s₃ \ s₁ ∪ s₁)
rw [diff_union_of_subset (h12.trans h23)]
no goals
α : Type u_1
β : Type ?u.27550
γ : Type ?u.27553
δ : Type ?u.27556
ι : Type ?u.27559
R : Type ?u.27562
R' : Type ?u.27565
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s s₁✝ s₂✝ t s₁ s₂ s₃ : Set α
h12 : s₁ ⊆ s₂
h23 : s₂ ⊆ s₃
h_nulldiff : ↑↑μ (s₃ \ s₁) = 0
le12 : ↑↑μ s₁ ≤ ↑↑μ s₂
le23 : ↑↑μ s₂ ≤ ↑↑μ s₃
⊢ ↑↑μ (s₃ \ s₁) + ↑↑μ s₁ = ↑↑μ s₁
simp only [h_nulldiff, zero_add]