Theorem: MeasureTheory.Measure.isTopologicalBasis_isOpen_lt_top
File: Mathlib/MeasureTheory/Measure/Typeclasses.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ✝ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
⊢ TopologicalSpace.IsTopologicalBasis {s | IsOpen s ∧ μ s < ⊤}
state_after: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ✝ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
⊢ ∀ (a : α) (u : Set α), a ∈ u → IsOpen u → ∃ v ∈ {s | IsOpen s ∧ μ s < ⊤}, a ∈ v ∧ v ⊆ u
tactic: refine TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds (fun s hs => hs.1) ?_
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ✝ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
⊢ ∀ (a : α) (u : Set α), a ∈ u → IsOpen u → ∃ v ∈ {s | IsOpen s ∧ μ s < ⊤}, a ∈ v ∧ v ⊆ u
state_after: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ✝ ν ν₁ ν₂ : Measure α
s✝ t : Set α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
x : α
s : Set α
xs : x ∈ s
hs : IsOpen s
⊢ ∃ v ∈ {s | IsOpen s ∧ μ s < ⊤}, x ∈ v ∧ v ⊆ s
tactic: intro x s xs hs
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ✝ ν ν₁ ν₂ : Measure α
s✝ t : Set α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
x : α
s : Set α
xs : x ∈ s
hs : IsOpen s
⊢ ∃ v ∈ {s | IsOpen s ∧ μ s < ⊤}, x ∈ v ∧ v ⊆ s
state_after: case intro.intro.intro
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ✝ ν ν₁ ν₂ : Measure α
s✝ t : Set α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
x : α
s : Set α
xs : x ∈ s
hs : IsOpen s
v : Set α
xv : x ∈ v
hv : IsOpen v
μv : μ v < ⊤
⊢ ∃ v ∈ {s | IsOpen s ∧ μ s < ⊤}, x ∈ v ∧ v ⊆ s
tactic: rcases μ.exists_isOpen_measure_lt_top x with ⟨v, xv, hv, μv⟩
state_before: case intro.intro.intro
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ✝ ν ν₁ ν₂ : Measure α
s✝ t : Set α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
x : α
s : Set α
xs : x ∈ s
hs : IsOpen s
v : Set α
xv : x ∈ v
hv : IsOpen v
μv : μ v < ⊤
⊢ ∃ v ∈ {s | IsOpen s ∧ μ s < ⊤}, x ∈ v ∧ v ⊆ s
state_after: case intro.intro.intro
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ✝ ν ν₁ ν₂ : Measure α
s✝ t : Set α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
x : α
s : Set α
xs : x ∈ s
hs : IsOpen s
v : Set α
xv : x ∈ v
hv : IsOpen v
μv : μ v < ⊤
⊢ μ (v ∩ s) ≤ μ v
tactic: refine ⟨v ∩ s, ⟨hv.inter hs, lt_of_le_of_lt ?_ μv⟩, ⟨xv, xs⟩, inter_subset_right⟩
state_before: case intro.intro.intro
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ✝ ν ν₁ ν₂ : Measure α
s✝ t : Set α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
x : α
s : Set α
xs : x ∈ s
hs : IsOpen s
v : Set α
xv : x ∈ v
hv : IsOpen v
μv : μ v < ⊤
⊢ μ (v ∩ s) ≤ μ v
state_after: no goals
tactic: exact measure_mono inter_subset_left