Theorem: MeasureTheory.Measure.InnerRegular.map
File: Mathlib/MeasureTheory/Measure/Regular.lean

tactic: intro U hU r hr
state_before: α✝ : Type ?u.3087
m : MeasurableSpace α✝
μ✝ : Measure α✝
p q : Set α✝ → Prop
U : Set α✝
ε : ℝ≥0∞
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
pa qa : Set α → Prop
H : InnerRegular μ pa qa
f : α ≃ β
hf : AEMeasurable ↑f
pb qb : Set β → Prop
hAB : ∀ (U : Set β), qb U → qa (↑f ⁻¹' U)
hAB' : ∀ (K : Set α), pa K → pb (↑f '' K)
hB₁ : ∀ (K : Set β), pb K → MeasurableSet K
hB₂ : ∀ (U : Set β), qb U → MeasurableSet U
⊢ InnerRegular (Measure.map (↑f) μ) pb qb

tactic: rw [map_apply_of_aemeasurable hf (hB₂ _ hU)] at hr
state_before: α✝ : Type ?u.3087
m : MeasurableSpace α✝
μ✝ : Measure α✝
p q : Set α✝ → Prop
U✝ : Set α✝
ε : ℝ≥0∞
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
pa qa : Set α → Prop
H : InnerRegular μ pa qa
f : α ≃ β
hf : AEMeasurable ↑f
pb qb : Set β → Prop
hAB : ∀ (U : Set β), qb U → qa (↑f ⁻¹' U)
hAB' : ∀ (K : Set α), pa K → pb (↑f '' K)
hB₁ : ∀ (K : Set β), pb K → MeasurableSet K
hB₂ : ∀ (U : Set β), qb U → MeasurableSet U
U : Set β
hU : qb U
r : ℝ≥0∞
hr : r < ↑↑(Measure.map (↑f) μ) U
⊢ ∃ K, K ⊆ U ∧ pb K ∧ r < ↑↑(Measure.map (↑f) μ) K

tactic: rcases H (hAB U hU) r hr with ⟨K, hKU, hKc, hK⟩
state_before: α✝ : Type ?u.3087
m : MeasurableSpace α✝
μ✝ : Measure α✝
p q : Set α✝ → Prop
U✝ : Set α✝
ε : ℝ≥0∞
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
pa qa : Set α → Prop
H : InnerRegular μ pa qa
f : α ≃ β
hf : AEMeasurable ↑f
pb qb : Set β → Prop
hAB : ∀ (U : Set β), qb U → qa (↑f ⁻¹' U)
hAB' : ∀ (K : Set α), pa K → pb (↑f '' K)
hB₁ : ∀ (K : Set β), pb K → MeasurableSet K
hB₂ : ∀ (U : Set β), qb U → MeasurableSet U
U : Set β
hU : qb U
r : ℝ≥0∞
hr : r < ↑↑μ (↑f ⁻¹' U)
⊢ ∃ K, K ⊆ U ∧ pb K ∧ r < ↑↑(Measure.map (↑f) μ) K

tactic: refine' ⟨f '' K, image_subset_iff.2 hKU, hAB' _ hKc, _⟩
state_before: case intro.intro.intro
α✝ : Type ?u.3087
m : MeasurableSpace α✝
μ✝ : Measure α✝
p q : Set α✝ → Prop
U✝ : Set α✝
ε : ℝ≥0∞
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
pa qa : Set α → Prop
H : InnerRegular μ pa qa
f : α ≃ β
hf : AEMeasurable ↑f
pb qb : Set β → Prop
hAB : ∀ (U : Set β), qb U → qa (↑f ⁻¹' U)
hAB' : ∀ (K : Set α), pa K → pb (↑f '' K)
hB₁ : ∀ (K : Set β), pb K → MeasurableSet K
hB₂ : ∀ (U : Set β), qb U → MeasurableSet U
U : Set β
hU : qb U
r : ℝ≥0∞
hr : r < ↑↑μ (↑f ⁻¹' U)
K : Set α
hKU : K ⊆ ↑f ⁻¹' U
hKc : pa K
hK : r < ↑↑μ K
⊢ ∃ K, K ⊆ U ∧ pb K ∧ r < ↑↑(Measure.map (↑f) μ) K

tactic: rwa [map_apply_of_aemeasurable hf (hB₁ _ <| hAB' _ hKc), f.preimage_image]
state_before: case intro.intro.intro
α✝ : Type ?u.3087
m : MeasurableSpace α✝
μ✝ : Measure α✝
p q : Set α✝ → Prop
U✝ : Set α✝
ε : ℝ≥0∞
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
pa qa : Set α → Prop
H : InnerRegular μ pa qa
f : α ≃ β
hf : AEMeasurable ↑f
pb qb : Set β → Prop
hAB : ∀ (U : Set β), qb U → qa (↑f ⁻¹' U)
hAB' : ∀ (K : Set α), pa K → pb (↑f '' K)
hB₁ : ∀ (K : Set β), pb K → MeasurableSet K
hB₂ : ∀ (U : Set β), qb U → MeasurableSet U
U : Set β
hU : qb U
r : ℝ≥0∞
hr : r < ↑↑μ (↑f ⁻¹' U)
K : Set α
hKU : K ⊆ ↑f ⁻¹' U
hKc : pa K
hK : r < ↑↑μ K
⊢ r < ↑↑(Measure.map (↑f) μ) (↑f '' K)