Theorem: MeasureTheory.Measure.MutuallySingular.sum_left
File: Mathlib/MeasureTheory/Measure/MutuallySingular.lean

tactic: refine ⟨fun h i => h.mono (le_sum _ _) le_rfl, fun H => ?_⟩
state_before: α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
⊢ sum μ ⟂ₘ ν ↔ ∀ (i : ι), μ i ⟂ₘ ν
state_after: α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
H : ∀ (i : ι), μ i ⟂ₘ ν
⊢ sum μ ⟂ₘ ν

tactic: choose s hsm hsμ hsν using H
state_before: α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
H : ∀ (i : ι), μ i ⟂ₘ ν
⊢ sum μ ⟂ₘ ν
state_after: α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
s : ι → Set α
hsm : ∀ (i : ι), MeasurableSet (s i)
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsν : ∀ (i : ι), ν (s i)ᶜ = 0
⊢ sum μ ⟂ₘ ν

tactic: refine ⟨⋂ i, s i, MeasurableSet.iInter hsm, ?_, ?_⟩
state_before: α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
s : ι → Set α
hsm : ∀ (i : ι), MeasurableSet (s i)
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsν : ∀ (i : ι), ν (s i)ᶜ = 0
⊢ sum μ ⟂ₘ ν
state_after: case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
s : ι → Set α
hsm : ∀ (i : ι), MeasurableSet (s i)
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsν : ∀ (i : ι), ν (s i)ᶜ = 0
⊢ (sum μ) (⋂ i, s i) = 0

case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
s : ι → Set α
hsm : ∀ (i : ι), MeasurableSet (s i)
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsν : ∀ (i : ι), ν (s i)ᶜ = 0
⊢ ν (⋂ i, s i)ᶜ = 0

tactic: rw [sum_apply _ (MeasurableSet.iInter hsm), ENNReal.tsum_eq_zero]
state_before: case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
s : ι → Set α
hsm : ∀ (i : ι), MeasurableSet (s i)
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsν : ∀ (i : ι), ν (s i)ᶜ = 0
⊢ (sum μ) (⋂ i, s i) = 0
state_after: case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
s : ι → Set α
hsm : ∀ (i : ι), MeasurableSet (s i)
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsν : ∀ (i : ι), ν (s i)ᶜ = 0
⊢ ∀ (i : ι), (μ i) (⋂ b, s b) = 0

tactic: exact fun i => measure_mono_null (iInter_subset _ _) (hsμ i)
state_before: case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
s : ι → Set α
hsm : ∀ (i : ι), MeasurableSet (s i)
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsν : ∀ (i : ι), ν (s i)ᶜ = 0
⊢ ∀ (i : ι), (μ i) (⋂ b, s b) = 0
state_after: no goals

tactic: rwa [compl_iInter, measure_iUnion_null_iff]
state_before: case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
s : ι → Set α
hsm : ∀ (i : ι), MeasurableSet (s i)
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsν : ∀ (i : ι), ν (s i)ᶜ = 0
⊢ ν (⋂ i, s i)ᶜ = 0
state_after: no goals