-- Mathlib/MeasureTheory/Group/Measure.lean
/-- Continuity of the measure of translates of a compact set: Given a compact set `k` in a
topological group, for `g` close enough to the origin, `μ (g • k \ k)` is arbitrarily small. -/
@[to_additive]
lemma eventually_nhds_one_measure_smul_diff_lt [LocallyCompactSpace G]
    [IsFiniteMeasureOnCompacts μ] [InnerRegularCompactLTTop μ] {k : Set G}
    (hk : IsCompact k) (h'k : IsClosed k) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∀ᶠ g in 𝓝 (1 : G), μ (g • k \ k) < ε

Proof state:
case intro.intro.intro
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : BorelSpace G
μ : Measure G
inst✝⁴ : Group G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
hk : IsCompact k
h'k : IsClosed k
ε : ℝ≥0∞
hε : ε ≠ 0
U : Set G
hUk : k ⊆ U
hU : IsOpen U
hμUk : μ U < μ k + ε
⊢ ∀ᶠ (g : G) in 𝓝 1, μ (g • k \ k) < ε

Tactic:
obtain ⟨V, hV1, hVkU⟩ : ∃ V ∈ 𝓝 (1 : G), V * k ⊆ U := compact_open_separated_mul_left hk hU hUk

Result:
case intro.intro.intro.intro.intro
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : BorelSpace G
μ : Measure G
inst✝⁴ : Group G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
hk : IsCompact k
h'k : IsClosed k
ε : ℝ≥0∞
hε : ε ≠ 0
U : Set G
hUk : k ⊆ U
hU : IsOpen U
hμUk : μ U < μ k + ε
V : Set G
hV1 : V ∈ 𝓝 1
hVkU : V * k ⊆ U
⊢ ∀ᶠ (g : G) in 𝓝 1, μ (g • k \ k) < ε

Proof state:
case h.h
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : BorelSpace G
μ : Measure G
inst✝⁴ : Group G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
hk : IsCompact k
h'k : IsClosed k
ε : ℝ≥0∞
hε : ε ≠ 0
U : Set G
hUk : k ⊆ U
hU : IsOpen U
hμUk : μ U < μ k + ε
V : Set G
hV1 : V ∈ 𝓝 1
hVkU : V * k ⊆ U
g : G
hg : g ∈ V
⊢ g • k ⊆ U

Tactic:
exact (smul_set_subset_smul hg).trans hVkU

Result:
no goals

Proof state:
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : BorelSpace G
μ : Measure G
inst✝⁴ : Group G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
hk : IsCompact k
h'k : IsClosed k
ε : ℝ≥0∞
hε : ε ≠ 0
U : Set G
hUk : k ⊆ U
hU : IsOpen U
hμUk : μ U < μ k + ε
V : Set G
hV1 : V ∈ 𝓝 1
hVkU : V * k ⊆ U
g : G
hg : g ∈ V
⊢ μ (g • k \ k) ≤ μ (U \ k)

Tactic:
gcongr

Result:
case h.h
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : BorelSpace G
μ : Measure G
inst✝⁴ : Group G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
hk : IsCompact k
h'k : IsClosed k
ε : ℝ≥0∞
hε : ε ≠ 0
U : Set G
hUk : k ⊆ U
hU : IsOpen U
hμUk : μ U < μ k + ε
V : Set G
hV1 : V ∈ 𝓝 1
hVkU : V * k ⊆ U
g : G
hg : g ∈ V
⊢ g • k ⊆ U

Proof state:
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : BorelSpace G
μ : Measure G
inst✝⁴ : Group G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
hk : IsCompact k
h'k : IsClosed k
ε : ℝ≥0∞
hε : ε ≠ 0
⊢ ∀ᶠ (g : G) in 𝓝 1, μ (g • k \ k) < ε

Tactic:
obtain ⟨U, hUk, hU, hμUk⟩ : ∃ (U : Set G), k ⊆ U ∧ IsOpen U ∧ μ U < μ k + ε :=
  hk.exists_isOpen_lt_add hε

Result:
case intro.intro.intro
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : BorelSpace G
μ : Measure G
inst✝⁴ : Group G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
hk : IsCompact k
h'k : IsClosed k
ε : ℝ≥0∞
hε : ε ≠ 0
U : Set G
hUk : k ⊆ U
hU : IsOpen U
hμUk : μ U < μ k + ε
⊢ ∀ᶠ (g : G) in 𝓝 1, μ (g • k \ k) < ε

Proof state:
case intro.intro.intro.intro.intro
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : BorelSpace G
μ : Measure G
inst✝⁴ : Group G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
hk : IsCompact k
h'k : IsClosed k
ε : ℝ≥0∞
hε : ε ≠ 0
U : Set G
hUk : k ⊆ U
hU : IsOpen U
hμUk : μ U < μ k + ε
V : Set G
hV1 : V ∈ 𝓝 1
hVkU : V * k ⊆ U
⊢ ∀ᶠ (g : G) in 𝓝 1, μ (g • k \ k) < ε

Tactic:
filter_upwards [hV1] with g hg

Result:
case h
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : BorelSpace G
μ : Measure G
inst✝⁴ : Group G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
hk : IsCompact k
h'k : IsClosed k
ε : ℝ≥0∞
hε : ε ≠ 0
U : Set G
hUk : k ⊆ U
hU : IsOpen U
hμUk : μ U < μ k + ε
V : Set G
hV1 : V ∈ 𝓝 1
hVkU : V * k ⊆ U
g : G
hg : g ∈ V
⊢ μ (g • k \ k) < ε

Proof state:
case h
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : BorelSpace G
μ : Measure G
inst✝⁴ : Group G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
hk : IsCompact k
h'k : IsClosed k
ε : ℝ≥0∞
hε : ε ≠ 0
U : Set G
hUk : k ⊆ U
hU : IsOpen U
hμUk : μ U < μ k + ε
V : Set G
hV1 : V ∈ 𝓝 1
hVkU : V * k ⊆ U
g : G
hg : g ∈ V
⊢ μ (g • k \ k) < ε

Tactic:
calc
  μ (g • k \ k) ≤ μ (U \ k) := by
    gcongr
    exact (smul_set_subset_smul hg).trans hVkU
  _ < ε := measure_diff_lt_of_lt_add h'k.nullMeasurableSet hUk hk.measure_lt_top.ne hμUk

Result:
no goals