-- Mathlib/MeasureTheory/Measure/Haar/Unique.lean
/-- Two left invariant measures give the same mass to level sets of continuous compactly supported
functions, up to the scalar `haarScalarFactor μ' μ`.

Auxiliary lemma in the proof of the more general
`measure_isMulInvariant_eq_smul_of_isCompact_closure`, which works for any set with
compact closure. -/
@[to_additive measure_preimage_isAddLeftInvariant_eq_smul_of_hasCompactSupport
"Two left invariant measures give the same mass to level sets of continuous compactly supported
functions, up to the scalar `addHaarScalarFactor μ' μ`.

Auxiliary lemma in the proof of the more general
`measure_isAddInvariant_eq_smul_of_isCompact_closure`, which works for any set with
compact closure."]
lemma measure_preimage_isMulLeftInvariant_eq_smul_of_hasCompactSupport
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    {f : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f) :
    μ' (f ⁻¹' {1}) = haarScalarFactor μ' μ • μ (f ⁻¹' {1})

Proof state:
case h.e'_5.h.e'_3
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
f : G → ℝ
hf : Continuous f
h'f : HasCompactSupport f
u : ℕ → ℝ
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → ℝ → ℝ := fun n x => ↑((thickenedIndicator ⋯ {1}) x)
vf_cont : ∀ (n : ℕ), Continuous (v n ∘ f)
ν : Measure G
hν : IsFiniteMeasureOnCompacts ν
x : G
T : Tendsto (fun i => (thickenedIndicatorAux (u i) {1} (f x)).toNNReal) atTop (𝓝 ({1}.indicator (fun x => 1) (f x)))
⊢ {1}.indicator (fun x => 1) (f x) = ↑({1}.indicator (fun x => 1) (f x))

Tactic:
simp

Result:
no goals

Proof state:
case neg
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
f : G → ℝ
hf : Continuous f
h'f : HasCompactSupport f
u : ℕ → ℝ
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → ℝ → ℝ := fun n x => ↑((thickenedIndicator ⋯ {1}) x)
vf_cont : ∀ (n : ℕ), Continuous (v n ∘ f)
ν : Measure G
hν : IsFiniteMeasureOnCompacts ν
n : ℕ
x : G
hx : x ∉ tsupport f
⊢ ↑((thickenedIndicator ⋯ {1}) (f x)) ≤ 0

Tactic:
rw [thickenedIndicator_zero]

Result:
case neg
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
f : G → ℝ
hf : Continuous f
h'f : HasCompactSupport f
u : ℕ → ℝ
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → ℝ → ℝ := fun n x => ↑((thickenedIndicator ⋯ {1}) x)
vf_cont : ∀ (n : ℕ), Continuous (v n ∘ f)
ν : Measure G
hν : IsFiniteMeasureOnCompacts ν
n : ℕ
x : G
hx : x ∉ tsupport f
⊢ ↑0 ≤ 0

case neg.x_out
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
f : G → ℝ
hf : Continuous f
h'f : HasCompactSupport f
u : ℕ → ℝ
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → ℝ → ℝ := fun n x => ↑((thickenedIndicator ⋯ {1}) x)
vf_cont : ∀ (n : ℕ), Continuous (v n ∘ f)
ν : Measure G
hν : IsFiniteMeasureOnCompacts ν
n : ℕ
x : G
hx : x ∉ tsupport f
⊢ f x ∉ Metric.thickening (u n) {1}

Proof state:
case neg.x_out
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
f : G → ℝ
hf : Continuous f
h'f : HasCompactSupport f
u : ℕ → ℝ
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → ℝ → ℝ := fun n x => ↑((thickenedIndicator ⋯ {1}) x)
vf_cont : ∀ (n : ℕ), Continuous (v n ∘ f)
ν : Measure G
hν : IsFiniteMeasureOnCompacts ν
n : ℕ
x : G
hx : x ∉ tsupport f
⊢ f x ∉ Metric.thickening (u n) {1}

Tactic:
simpa [image_eq_zero_of_nmem_tsupport hx] using (u_mem n).2.le

Result:
no goals

Proof state:
case h_lim.h
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
f : G → ℝ
hf : Continuous f
h'f : HasCompactSupport f
u : ℕ → ℝ
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → ℝ → ℝ := fun n x => ↑((thickenedIndicator ⋯ {1}) x)
vf_cont : ∀ (n : ℕ), Continuous (v n ∘ f)
ν : Measure G
hν : IsFiniteMeasureOnCompacts ν
x : G
T : Tendsto (fun i => (thickenedIndicatorAux (u i) {1} (f x)).toNNReal) atTop (𝓝 ({1}.indicator (fun x => 1) (f x)))
⊢ Tendsto (fun n => v n (f x)) atTop (𝓝 ({1}.indicator (fun x => 1) (f x)))

Tactic:
convert NNReal.tendsto_coe.2 T

Result:
case h.e'_5.h.e'_3
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
f : G → ℝ
hf : Continuous f
h'f : HasCompactSupport f
u : ℕ → ℝ
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → ℝ → ℝ := fun n x => ↑((thickenedIndicator ⋯ {1}) x)
vf_cont : ∀ (n : ℕ), Continuous (v n ∘ f)
ν : Measure G
hν : IsFiniteMeasureOnCompacts ν
x : G
T : Tendsto (fun i => (thickenedIndicatorAux (u i) {1} (f x)).toNNReal) atTop (𝓝 ({1}.indicator (fun x => 1) (f x)))
⊢ {1}.indicator (fun x => 1) (f x) = ↑({1}.indicator (fun x => 1) (f x))

Proof state:
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
f : G → ℝ
hf : Continuous f
h'f : HasCompactSupport f
u : ℕ → ℝ
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → ℝ → ℝ := fun n x => ↑((thickenedIndicator ⋯ {1}) x)
vf_cont : ∀ (n : ℕ), Continuous (v n ∘ f)
I :
  ∀ (ν : Measure G),
    IsFiniteMeasureOnCompacts ν →
      Tendsto (fun n => ∫ (x : G), v n (f x) ∂ν) atTop (𝓝 (∫ (x : G), {1}.indicator (fun x => 1) (f x) ∂ν))
M : ∀ (n : ℕ), ∫ (x : G), v n (f x) ∂μ' = ∫ (x : G), v n (f x) ∂μ'.haarScalarFactor μ • μ
I1 :
  Tendsto (fun n => ∫ (x : G), v n (f x) ∂μ'.haarScalarFactor μ • μ) atTop
    (𝓝 (∫ (x : G), {1}.indicator (fun x => 1) (f x) ∂μ'))
⊢ IsFiniteMeasureOnCompacts (μ'.haarScalarFactor μ • μ)

Tactic:
infer_instance

Result:
no goals