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[Merged by Bors] - chore: forward-port leanprover-community/mathlib#18983 #4001

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[Merged by Bors] - chore: forward-port leanprover-community/mathlib#18983 #4001

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8b918ff
mathport
Ruben-VandeVelde May 15, 2023
4b50629
Fix
Ruben-VandeVelde May 15, 2023
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17 changes: 16 additions & 1 deletion Mathlib/MeasureTheory/Measure/MeasureSpace.lean
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Expand Up @@ -4,7 +4,7 @@
Authors: Johannes Hölzl, Mario Carneiro

! This file was ported from Lean 3 source module measure_theory.measure.measure_space
! leanprover-community/mathlib commit 88fcb83fe7996142dfcfe7368d31304a9adc874a
! leanprover-community/mathlib commit 343e80208d29d2d15f8050b929aa50fe4ce71b55

Check notice on line 7 in Mathlib/MeasureTheory/Measure/MeasureSpace.lean

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Synchronization 

See https://leanprover-community.github.io/mathlib-port-status/file/measure_theory/measure/measure_space?range=88fcb83fe7996142dfcfe7368d31304a9adc874a..343e80208d29d2d15f8050b929aa50fe4ce71b55
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
Expand Down Expand Up @@ -1058,6 +1058,21 @@

end sInf

@[simp]
theorem _root_.MeasureTheory.OuterMeasure.toMeasure_top [MeasurableSpace α] :
(⊤ : OuterMeasure α).toMeasure (by rw [OuterMeasure.top_caratheodory]; exact le_top) =
(⊤ : Measure α) :=
top_unique fun s hs => by
cases' s.eq_empty_or_nonempty with h h <;>
simp [h, toMeasure_apply ⊤ _ hs, OuterMeasure.top_apply]
#align measure_theory.outer_measure.to_measure_top MeasureTheory.OuterMeasure.toMeasure_top

@[simp]
theorem toOuterMeasure_top [MeasurableSpace α] :
(⊤ : Measure α).toOuterMeasure = (⊤ : OuterMeasure α) := by
rw [← OuterMeasure.toMeasure_top, toMeasure_toOuterMeasure, OuterMeasure.trim_top]
#align measure_theory.measure.to_outer_measure_top MeasureTheory.Measure.toOuterMeasure_top

@[simp]
theorem top_add : ⊤ + μ = ⊤ :=
top_unique <| Measure.le_add_right le_rfl
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26 changes: 25 additions & 1 deletion Mathlib/MeasureTheory/Measure/OuterMeasure.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -4,7 +4,7 @@
Authors: Johannes Hölzl, Mario Carneiro

! This file was ported from Lean 3 source module measure_theory.measure.outer_measure
! leanprover-community/mathlib commit ec4b2eeb50364487f80421c0b4c41328a611f30d
! leanprover-community/mathlib commit 343e80208d29d2d15f8050b929aa50fe4ce71b55

Check notice on line 7 in Mathlib/MeasureTheory/Measure/OuterMeasure.lean

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Synchronization 

See https://leanprover-community.github.io/mathlib-port-status/file/measure_theory/measure/outer_measure?range=ec4b2eeb50364487f80421c0b4c41328a611f30d..343e80208d29d2d15f8050b929aa50fe4ce71b55
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
Expand Down Expand Up @@ -882,6 +882,20 @@
cases' s.eq_empty_or_nonempty with h h <;> simp [h]
#align measure_theory.outer_measure.le_bounded_by' MeasureTheory.OuterMeasure.le_bounded_by'

@[simp]
theorem boundedBy_top : boundedBy (⊤ : Set α → ℝ≥0∞) = ⊤ := by
rw [eq_top_iff, le_bounded_by']
intro s hs
rw [top_apply hs]
exact le_rfl
#align measure_theory.outer_measure.bounded_by_top MeasureTheory.OuterMeasure.boundedBy_top

@[simp]
theorem boundedBy_zero : boundedBy (0 : Set α → ℝ≥0∞) = 0 := by
rw [← coe_bot, eq_bot_iff]
apply boundedBy_le
#align measure_theory.outer_measure.bounded_by_zero MeasureTheory.OuterMeasure.boundedBy_zero

theorem smul_boundedBy {c : ℝ≥0∞} (hc : c ≠ ∞) : c • boundedBy m = boundedBy (c • m) := by
simp only [boundedBy , smul_ofFunction hc]
congr 1 with s : 1
Expand Down Expand Up @@ -1334,6 +1348,11 @@
iInf_congr_Prop hP fun _h => hm _ _
#align measure_theory.extend_congr MeasureTheory.extend_congr

@[simp]
theorem extend_top {α : Type _} {P : α → Prop} : extend (fun _ _ => ∞ : ∀ s : α, P s → ℝ≥0∞) = ⊤ :=
funext fun _ => iInf_eq_top.mpr fun _ => rfl
#align measure_theory.extend_top MeasureTheory.extend_top

end Extend

section ExtendSet
Expand Down Expand Up @@ -1648,6 +1667,11 @@
trim_eq_trim_iff.2 fun _s => m.trim_eq
#align measure_theory.outer_measure.trim_trim MeasureTheory.OuterMeasure.trim_trim

@[simp]
theorem trim_top : (⊤ : OuterMeasure α).trim = ⊤ :=
eq_top_iff.2 <| le_trim _
#align measure_theory.outer_measure.trim_top MeasureTheory.OuterMeasure.trim_top

@[simp]
theorem trim_zero : (0 : OuterMeasure α).trim = 0 :=
ext fun s =>
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