Theorem: MeasureTheory.Measure.InnerRegularCompactLTTop.map_of_continuous
File: Mathlib/MeasureTheory/Measure/Regular.lean

tactic: constructor
state_before: α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
h : μ.InnerRegularCompactLTTop
f : α → β
hf : Continuous f
⊢ (map f μ).InnerRegularCompactLTTop
state_after: case innerRegular
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
h : μ.InnerRegularCompactLTTop
f : α → β
hf : Continuous f
⊢ (map f μ).InnerRegularWRT IsCompact fun s => MeasurableSet s ∧ (map f μ) s ≠ ⊤

tactic: refine InnerRegularWRT.map h.innerRegular hf.aemeasurable ?_ (fun K hK ↦ hK.image hf) ?_
state_before: case innerRegular
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
h : μ.InnerRegularCompactLTTop
f : α → β
hf : Continuous f
⊢ (map f μ).InnerRegularWRT IsCompact fun s => MeasurableSet s ∧ (map f μ) s ≠ ⊤
state_after: case innerRegular.refine_1
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
h : μ.InnerRegularCompactLTTop
f : α → β
hf : Continuous f
⊢ ∀ (U : Set β), MeasurableSet U ∧ (map f μ) U ≠ ⊤ → MeasurableSet (f ⁻¹' U) ∧ μ (f ⁻¹' U) ≠ ⊤

case innerRegular.refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
h : μ.InnerRegularCompactLTTop
f : α → β
hf : Continuous f
⊢ ∀ (U : Set β), MeasurableSet U ∧ (map f μ) U ≠ ⊤ → MeasurableSet U

tactic: rintro s ⟨hs, h's⟩
state_before: case innerRegular.refine_1
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
h : μ.InnerRegularCompactLTTop
f : α → β
hf : Continuous f
⊢ ∀ (U : Set β), MeasurableSet U ∧ (map f μ) U ≠ ⊤ → MeasurableSet (f ⁻¹' U) ∧ μ (f ⁻¹' U) ≠ ⊤
state_after: case innerRegular.refine_1.intro
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
h : μ.InnerRegularCompactLTTop
f : α → β
hf : Continuous f
s : Set β
hs : MeasurableSet s
h's : (map f μ) s ≠ ⊤
⊢ MeasurableSet (f ⁻¹' s) ∧ μ (f ⁻¹' s) ≠ ⊤

tactic: exact ⟨hf.measurable hs, by rwa [map_apply hf.measurable hs] at h's⟩
state_before: case innerRegular.refine_1.intro
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
h : μ.InnerRegularCompactLTTop
f : α → β
hf : Continuous f
s : Set β
hs : MeasurableSet s
h's : (map f μ) s ≠ ⊤
⊢ MeasurableSet (f ⁻¹' s) ∧ μ (f ⁻¹' s) ≠ ⊤
state_after: no goals

tactic: rwa [map_apply hf.measurable hs] at h's
state_before: α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
h : μ.InnerRegularCompactLTTop
f : α → β
hf : Continuous f
s : Set β
hs : MeasurableSet s
h's : (map f μ) s ≠ ⊤
⊢ μ (f ⁻¹' s) ≠ ⊤
state_after: no goals

tactic: rintro s ⟨hs, -⟩
state_before: case innerRegular.refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
h : μ.InnerRegularCompactLTTop
f : α → β
hf : Continuous f
⊢ ∀ (U : Set β), MeasurableSet U ∧ (map f μ) U ≠ ⊤ → MeasurableSet U
state_after: case innerRegular.refine_2.intro
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
h : μ.InnerRegularCompactLTTop
f : α → β
hf : Continuous f
s : Set β
hs : MeasurableSet s
⊢ MeasurableSet s

tactic: exact hs
state_before: case innerRegular.refine_2.intro
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
h : μ.InnerRegularCompactLTTop
f : α → β
hf : Continuous f
s : Set β
hs : MeasurableSet s
⊢ MeasurableSet s
state_after: no goals