Theorem: MeasureTheory.Measure.measure_eq_zero_of_subset_diff_everywherePosSubset
File: Mathlib/MeasureTheory/Measure/EverywherePos.lean

tactic: apply hk.induction_on (p := fun t ↦ μ t = 0)
state_before: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ μ k = 0
state_after: case he
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ μ ∅ = 0

case hmono
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ ⦃s t : Set α⦄, s ⊆ t → μ t = 0 → μ s = 0

case hunion
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ ⦃s t : Set α⦄, μ s = 0 → μ t = 0 → μ (s ∪ t) = 0

case hnhds
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ x ∈ k, ∃ t ∈ 𝓝[k] x, μ t = 0

tactic: exact measure_empty
state_before: case he
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ μ ∅ = 0
state_after: no goals

tactic: exact fun s t hst ht ↦ measure_mono_null hst ht
state_before: case hmono
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ ⦃s t : Set α⦄, s ⊆ t → μ t = 0 → μ s = 0
state_after: no goals

tactic: exact fun s t hs ht ↦ measure_union_null hs ht
state_before: case hunion
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ ⦃s t : Set α⦄, μ s = 0 → μ t = 0 → μ (s ∪ t) = 0
state_after: no goals

tactic: intro x hx
state_before: case hnhds
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ x ∈ k, ∃ t ∈ 𝓝[k] x, μ t = 0
state_after: case hnhds
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
x : α
hx : x ∈ k
⊢ ∃ t ∈ 𝓝[k] x, μ t = 0

tactic: obtain ⟨u, ux, hu⟩ : ∃ u ∈ 𝓝[s] x, μ u = 0 := by
  simpa [everywherePosSubset, (h'k hx).1] using (h'k hx).2
state_before: case hnhds
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
x : α
hx : x ∈ k
⊢ ∃ t ∈ 𝓝[k] x, μ t = 0
state_after: case hnhds.intro.intro
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
x : α
hx : x ∈ k
u : Set α
ux : u ∈ 𝓝[s] x
hu : μ u = 0
⊢ ∃ t ∈ 𝓝[k] x, μ t = 0

tactic: exact ⟨u, nhdsWithin_mono x (h'k.trans diff_subset) ux, hu⟩
state_before: case hnhds.intro.intro
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
x : α
hx : x ∈ k
u : Set α
ux : u ∈ 𝓝[s] x
hu : μ u = 0
⊢ ∃ t ∈ 𝓝[k] x, μ t = 0
state_after: no goals

tactic: simpa [everywherePosSubset, (h'k hx).1] using (h'k hx).2
state_before: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
x : α
hx : x ∈ k
⊢ ∃ u ∈ 𝓝[s] x, μ u = 0
state_after: no goals