-- Mathlib/Analysis/Convex/Measure.lean
theorem addHaar_frontier (hs : Convex ℝ s) : μ (frontier s) = 0

Proof state:
case inr.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s✝ : Set E
hs✝ : Convex ℝ s✝
x : E
hx✝ : x ∈ interior s✝
s : Set E
hs : Convex ℝ s
hx : x ∈ interior s
hb : μ (interior s) ≠ ⊤
d : ℕ := finrank ℝ E
this✝ : ∀ (r : ℝ≥0), 1 < r → μ (closure s) ≤ ↑(r ^ d) * μ (interior s)
this : ∀ᶠ (r : ℝ≥0) in 𝓝[>] 1, μ (closure s) ≤ ↑(r ^ d) * μ (interior s)
⊢ ((fun x => x * μ (interior s)) ∘ ENNReal.ofNNReal ∘ fun a => a ^ d) 1 = μ (interior s)

Tactic:
simp

Result:
no goals

Proof state:
case inr.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s✝ : Set E
hs✝ : Convex ℝ s✝
x : E
hx✝ : x ∈ interior s✝
s : Set E
hs : Convex ℝ s
hx : x ∈ interior s
hb : Bornology.IsBounded s
⊢ μ (frontier s) = 0

Tactic:
replace hb : μ (interior s) ≠ ∞ := (hb.subset interior_subset).measure_lt_top.ne

Result:
case inr.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s✝ : Set E
hs✝ : Convex ℝ s✝
x : E
hx✝ : x ∈ interior s✝
s : Set E
hs : Convex ℝ s
hx : x ∈ interior s
hb : μ (interior s) ≠ ⊤
⊢ μ (frontier s) = 0

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s✝ : Set E
hs✝ : Convex ℝ s✝
x : E
hx✝ : x ∈ interior s✝
s : Set E
hs : Convex ℝ s
hx : x ∈ interior s
hb : μ (interior s) ≠ ⊤
d : ℕ := finrank ℝ E
r : ℝ≥0
hr : 1 < r
⊢ μ (⇑(AffineMap.homothety x ↑r) '' interior s) = ↑(r ^ d) * μ (interior s)

Tactic:
rw [addHaar_image_homothety, ← NNReal.coe_pow, NNReal.abs_eq, ENNReal.ofReal_coe_nnreal]

Result:
no goals

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : Convex ℝ s
x : E
hx : x ∈ interior s
H : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Bornology.IsBounded t → μ (frontier t) = 0
B : ℕ → Set E := fun n => ball x (↑n + 1)
y : E
hy : y ∈ frontier s
N : ℕ := ⌊dist y x⌋₊
hN : y ∈ B N
⊢ y ∈ frontier s ∩ ball x (↑N + 1)

Tactic:
exact ⟨hy, hN⟩

Result:
no goals

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : Convex ℝ s
x : E
hx : x ∈ interior s
H : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Bornology.IsBounded t → μ (frontier t) = 0
B : ℕ → Set E := fun n => ball x (↑n + 1)
n : ℕ
⊢ x ∈ interior (s ∩ B n)

Tactic:
rw [interior_inter, isOpen_ball.interior_eq]

Result:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : Convex ℝ s
x : E
hx : x ∈ interior s
H : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Bornology.IsBounded t → μ (frontier t) = 0
B : ℕ → Set E := fun n => ball x (↑n + 1)
n : ℕ
⊢ x ∈ interior s ∩ ball x (↑n + 1)

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : Convex ℝ s
x : E
hx : x ∈ interior s
H : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Bornology.IsBounded t → μ (frontier t) = 0
B : ℕ → Set E := fun n => ball x (↑n + 1)
⊢ μ (⋃ n, frontier (s ∩ B n)) = 0

Tactic:
refine measure_iUnion_null fun n =>
  H _ (hs.inter (convex_ball _ _)) ?_ (isBounded_ball.subset inter_subset_right)

Result:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : Convex ℝ s
x : E
hx : x ∈ interior s
H : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Bornology.IsBounded t → μ (frontier t) = 0
B : ℕ → Set E := fun n => ball x (↑n + 1)
n : ℕ
⊢ x ∈ interior (s ∩ B n)

Proof state:
case inr.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : Convex ℝ s
x : E
hx : x ∈ interior s
⊢ ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Bornology.IsBounded t → μ (frontier t) = 0

Tactic:
intro s hs hx hb

Result:
case inr.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s✝ : Set E
hs✝ : Convex ℝ s✝
x : E
hx✝ : x ∈ interior s✝
s : Set E
hs : Convex ℝ s
hx : x ∈ interior s
hb : Bornology.IsBounded s
⊢ μ (frontier s) = 0

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : Convex ℝ s
x : E
hx : x ∈ interior s
H : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Bornology.IsBounded t → μ (frontier t) = 0
⊢ μ (frontier s) = 0

Tactic:
let B : ℕ → Set E := fun n => ball x (n + 1)

Result:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : Convex ℝ s
x : E
hx : x ∈ interior s
H : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Bornology.IsBounded t → μ (frontier t) = 0
B : ℕ → Set E := fun n => ball x (↑n + 1)
⊢ μ (frontier s) = 0

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : Convex ℝ s
x : E
hx : x ∈ interior s
H : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Bornology.IsBounded t → μ (frontier t) = 0
B : ℕ → Set E := fun n => ball x (↑n + 1)
y : E
hy : y ∈ frontier s
⊢ y ∈ ⋃ n, frontier (s ∩ B n)

Tactic:
set N : ℕ := ⌊dist y x⌋₊

Result:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : Convex ℝ s
x : E
hx : x ∈ interior s
H : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Bornology.IsBounded t → μ (frontier t) = 0
B : ℕ → Set E := fun n => ball x (↑n + 1)
y : E
hy : y ∈ frontier s
N : ℕ := ⌊dist y x⌋₊
⊢ y ∈ ⋃ n, frontier (s ∩ B n)