Theorem: ContinuousOn.integrableOn_compact'
File: Mathlib/MeasureTheory/Function/LocallyIntegrable.lean

Proof steps:
state_before: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝¹ : OpensMeasurableSpace X
K : Set X
a b : X
inst✝ : IsFiniteMeasureOnCompacts μ
hK : IsCompact K
h'K : MeasurableSet K
hf : ContinuousOn f K
⊢ IntegrableOn f K μ
state_after: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝¹ : OpensMeasurableSpace X
K : Set X
a b : X
inst✝ : IsFiniteMeasureOnCompacts μ
hK : IsCompact K
h'K : MeasurableSet K
hf : ContinuousOn f K
⊢ HasFiniteIntegral f (μ.restrict K)
tactic: refine ⟨ContinuousOn.aestronglyMeasurable_of_isCompact hf hK h'K, ?_⟩
state_before: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝¹ : OpensMeasurableSpace X
K : Set X
a b : X
inst✝ : IsFiniteMeasureOnCompacts μ
hK : IsCompact K
h'K : MeasurableSet K
hf : ContinuousOn f K
⊢ HasFiniteIntegral f (μ.restrict K)
state_after: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝¹ : OpensMeasurableSpace X
K : Set X
a b : X
inst✝ : IsFiniteMeasureOnCompacts μ
hK : IsCompact K
h'K : MeasurableSet K
hf : ContinuousOn f K
this : Fact (μ K < ⊤)
⊢ HasFiniteIntegral f (μ.restrict K)
tactic: have : Fact (μ K < ∞) := ⟨hK.measure_lt_top⟩
state_before: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝¹ : OpensMeasurableSpace X
K : Set X
a b : X
inst✝ : IsFiniteMeasureOnCompacts μ
hK : IsCompact K
h'K : MeasurableSet K
hf : ContinuousOn f K
this : Fact (μ K < ⊤)
⊢ HasFiniteIntegral f (μ.restrict K)
state_after: case intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝¹ : OpensMeasurableSpace X
K : Set X
a b : X
inst✝ : IsFiniteMeasureOnCompacts μ
hK : IsCompact K
h'K : MeasurableSet K
hf : ContinuousOn f K
this : Fact (μ K < ⊤)
C : ℝ
hC : ∀ x ∈ f '' K, ‖x‖ ≤ C
⊢ HasFiniteIntegral f (μ.restrict K)
tactic: obtain ⟨C, hC⟩ : ∃ C, ∀ x ∈ f '' K, ‖x‖ ≤ C :=
    IsBounded.exists_norm_le (hK.image_of_continuousOn hf).isBounded
state_before: case intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝¹ : OpensMeasurableSpace X
K : Set X
a b : X
inst✝ : IsFiniteMeasureOnCompacts μ
hK : IsCompact K
h'K : MeasurableSet K
hf : ContinuousOn f K
this : Fact (μ K < ⊤)
C : ℝ
hC : ∀ x ∈ f '' K, ‖x‖ ≤ C
⊢ HasFiniteIntegral f (μ.restrict K)
state_after: case intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝¹ : OpensMeasurableSpace X
K : Set X
a b : X
inst✝ : IsFiniteMeasureOnCompacts μ
hK : IsCompact K
h'K : MeasurableSet K
hf : ContinuousOn f K
this : Fact (μ K < ⊤)
C : ℝ
hC : ∀ x ∈ f '' K, ‖x‖ ≤ C
⊢ ∀ᵐ (a : X) ∂μ.restrict K, ‖f a‖ ≤ C
tactic: apply hasFiniteIntegral_of_bounded (C := C)
state_before: case intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝¹ : OpensMeasurableSpace X
K : Set X
a b : X
inst✝ : IsFiniteMeasureOnCompacts μ
hK : IsCompact K
h'K : MeasurableSet K
hf : ContinuousOn f K
this : Fact (μ K < ⊤)
C : ℝ
hC : ∀ x ∈ f '' K, ‖x‖ ≤ C
⊢ ∀ᵐ (a : X) ∂μ.restrict K, ‖f a‖ ≤ C
state_after: no goals
tactic: filter_upwards [ae_restrict_mem h'K] with x hx using hC _ (mem_image_of_mem f hx)