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

tactic: intro x hx n hn
state_before: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ ν : Measure α
s k : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
⊢ μ.IsEverywherePos (μ.everywherePosSubset s)
state_after: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ ν : Measure α
s k : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
⊢ 0 < μ n

tactic: rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 hn with ⟨u, u_mem, hu⟩
state_before: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ ν : Measure α
s k : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
⊢ 0 < μ n
state_after: case intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ ν : Measure α
s k : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
u : Set α
u_mem : u ∈ 𝓝 x
hu : u ∩ μ.everywherePosSubset s ⊆ n
⊢ 0 < μ n

tactic: have A : 0 < μ (u ∩ s) := by
  have : u ∩ s ∈ 𝓝[s] x := by rw [inter_comm]; exact inter_mem_nhdsWithin s u_mem
  exact hx.2 _ this
state_before: case intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ ν : Measure α
s k : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
u : Set α
u_mem : u ∈ 𝓝 x
hu : u ∩ μ.everywherePosSubset s ⊆ n
⊢ 0 < μ n
state_after: case intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ ν : Measure α
s k : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
u : Set α
u_mem : u ∈ 𝓝 x
hu : u ∩ μ.everywherePosSubset s ⊆ n
A : 0 < μ (u ∩ s)
⊢ 0 < μ n

tactic: have B : (u ∩ μ.everywherePosSubset s : Set α) =ᵐ[μ] (u ∩ s : Set α) :=
  ae_eq_set_inter (ae_eq_refl _) (everywherePosSubset_ae_eq hs)
state_before: case intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ ν : Measure α
s k : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
u : Set α
u_mem : u ∈ 𝓝 x
hu : u ∩ μ.everywherePosSubset s ⊆ n
A : 0 < μ (u ∩ s)
⊢ 0 < μ n
state_after: case intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ ν : Measure α
s k : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
u : Set α
u_mem : u ∈ 𝓝 x
hu : u ∩ μ.everywherePosSubset s ⊆ n
A : 0 < μ (u ∩ s)
B : u ∩ μ.everywherePosSubset s =ᶠ[ae μ] u ∩ s
⊢ 0 < μ n

tactic: rw [← B.measure_eq] at A
state_before: case intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ ν : Measure α
s k : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
u : Set α
u_mem : u ∈ 𝓝 x
hu : u ∩ μ.everywherePosSubset s ⊆ n
A : 0 < μ (u ∩ s)
B : u ∩ μ.everywherePosSubset s =ᶠ[ae μ] u ∩ s
⊢ 0 < μ n
state_after: case intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ ν : Measure α
s k : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
u : Set α
u_mem : u ∈ 𝓝 x
hu : u ∩ μ.everywherePosSubset s ⊆ n
A : 0 < μ (u ∩ μ.everywherePosSubset s)
B : u ∩ μ.everywherePosSubset s =ᶠ[ae μ] u ∩ s
⊢ 0 < μ n

tactic: exact A.trans_le (measure_mono hu)
state_before: case intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ ν : Measure α
s k : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
u : Set α
u_mem : u ∈ 𝓝 x
hu : u ∩ μ.everywherePosSubset s ⊆ n
A : 0 < μ (u ∩ μ.everywherePosSubset s)
B : u ∩ μ.everywherePosSubset s =ᶠ[ae μ] u ∩ s
⊢ 0 < μ n
state_after: no goals

tactic: have : u ∩ s ∈ 𝓝[s] x := by rw [inter_comm]; exact inter_mem_nhdsWithin s u_mem
state_before: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ ν : Measure α
s k : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
u : Set α
u_mem : u ∈ 𝓝 x
hu : u ∩ μ.everywherePosSubset s ⊆ n
⊢ 0 < μ (u ∩ s)
state_after: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ ν : Measure α
s k : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
u : Set α
u_mem : u ∈ 𝓝 x
hu : u ∩ μ.everywherePosSubset s ⊆ n
this : u ∩ s ∈ 𝓝[s] x
⊢ 0 < μ (u ∩ s)

tactic: exact hx.2 _ this
state_before: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ ν : Measure α
s k : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
u : Set α
u_mem : u ∈ 𝓝 x
hu : u ∩ μ.everywherePosSubset s ⊆ n
this : u ∩ s ∈ 𝓝[s] x
⊢ 0 < μ (u ∩ s)
state_after: no goals

tactic: rw [inter_comm]
state_before: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ ν : Measure α
s k : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
u : Set α
u_mem : u ∈ 𝓝 x
hu : u ∩ μ.everywherePosSubset s ⊆ n
⊢ u ∩ s ∈ 𝓝[s] x
state_after: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ ν : Measure α
s k : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
u : Set α
u_mem : u ∈ 𝓝 x
hu : u ∩ μ.everywherePosSubset s ⊆ n
⊢ s ∩ u ∈ 𝓝[s] x

tactic: exact inter_mem_nhdsWithin s u_mem
state_before: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ ν : Measure α
s k : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
u : Set α
u_mem : u ∈ 𝓝 x
hu : u ∩ μ.everywherePosSubset s ⊆ n
⊢ s ∩ u ∈ 𝓝[s] x
state_after: no goals