Theorem: MeasureTheory.LocallyIntegrable.integrableOn_nhds_isCompact
File: Mathlib/MeasureTheory/Function/LocallyIntegrable.lean

tactic: refine' IsCompact.induction_on hk _ _ _ _
state_before: X : Type u_1
Y : Type ?u.125788
E : Type u_2
R : Type ?u.125794
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
⊢ ∃ u, IsOpen u ∧ k ⊆ u ∧ IntegrableOn f u

tactic: refine' ⟨∅, isOpen_empty, Subset.rfl, integrableOn_empty⟩
state_before: case refine'_1
X : Type u_1
Y : Type ?u.125788
E : Type u_2
R : Type ?u.125794
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
⊢ ∃ u, IsOpen u ∧ ∅ ⊆ u ∧ IntegrableOn f u

tactic: rintro s t hst ⟨u, u_open, tu, hu⟩
state_before: case refine'_2
X : Type u_1
Y : Type ?u.125788
E : Type u_2
R : Type ?u.125794
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
⊢ ∀ ⦃s t : Set X⦄, s ⊆ t → (∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u) → ∃ u, IsOpen u ∧ s ⊆ u ∧ IntegrableOn f u

tactic: exact ⟨u, u_open, hst.trans tu, hu⟩
state_before: case refine'_2.intro.intro.intro
X : Type u_1
Y : Type ?u.125788
E : Type u_2
R : Type ?u.125794
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s✝ : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
s t : Set X
hst : s ⊆ t
u : Set X
u_open : IsOpen u
tu : t ⊆ u
hu : IntegrableOn f u
⊢ ∃ u, IsOpen u ∧ s ⊆ u ∧ IntegrableOn f u

tactic: rintro s t ⟨u, u_open, su, hu⟩ ⟨v, v_open, tv, hv⟩
state_before: case refine'_3
X : Type u_1
Y : Type ?u.125788
E : Type u_2
R : Type ?u.125794
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
⊢ ∀ ⦃s t : Set X⦄,
    (∃ u, IsOpen u ∧ s ⊆ u ∧ IntegrableOn f u) →
      (∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u) → ∃ u, IsOpen u ∧ s ∪ t ⊆ u ∧ IntegrableOn f u

tactic: exact ⟨u ∪ v, u_open.union v_open, union_subset_union su tv, hu.union hv⟩
state_before: case refine'_3.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type ?u.125788
E : Type u_2
R : Type ?u.125794
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s✝ : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
s t u : Set X
u_open : IsOpen u
su : s ⊆ u
hu : IntegrableOn f u
v : Set X
v_open : IsOpen v
tv : t ⊆ v
hv : IntegrableOn f v
⊢ ∃ u, IsOpen u ∧ s ∪ t ⊆ u ∧ IntegrableOn f u

tactic: intro x _
state_before: case refine'_4
X : Type u_1
Y : Type ?u.125788
E : Type u_2
R : Type ?u.125794
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
⊢ ∀ (x : X), x ∈ k → ∃ t, t ∈ 𝓝[k] x ∧ ∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u

tactic: rcases hf x with ⟨u, ux, hu⟩
state_before: case refine'_4
X : Type u_1
Y : Type ?u.125788
E : Type u_2
R : Type ?u.125794
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
x : X
a✝ : x ∈ k
⊢ ∃ t, t ∈ 𝓝[k] x ∧ ∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u

tactic: rcases mem_nhds_iff.1 ux with ⟨v, vu, v_open, xv⟩
state_before: case refine'_4.intro.intro
X : Type u_1
Y : Type ?u.125788
E : Type u_2
R : Type ?u.125794
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
x : X
a✝ : x ∈ k
u : Set X
ux : u ∈ 𝓝 x
hu : IntegrableOn f u
⊢ ∃ t, t ∈ 𝓝[k] x ∧ ∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u

tactic: exact ⟨v, nhdsWithin_le_nhds (v_open.mem_nhds xv), v, v_open, Subset.rfl, hu.mono_set vu⟩
state_before: case refine'_4.intro.intro.intro.intro.intro
X : Type u_1
Y : Type ?u.125788
E : Type u_2
R : Type ?u.125794
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
x : X
a✝ : x ∈ k
u : Set X
ux : u ∈ 𝓝 x
hu : IntegrableOn f u
v : Set X
vu : v ⊆ u
v_open : IsOpen v
xv : x ∈ v
⊢ ∃ t, t ∈ 𝓝[k] x ∧ ∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u