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Add massOn empty simp lemma #4

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Merged
Xinze-Li-Moqian merged 1 commit into from
May 13, 2026
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Add massOn empty simp lemma #4

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6 changes: 6 additions & 0 deletions OpenGALib/GeometricMeasureTheory/Varifold.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -123,6 +123,11 @@ omit [MeasureTheory.MeasureSpace M] in
theorem massOn_nonneg (V : Varifold M) (U : Set M) : 0 ≤ massOn V U :=
ENNReal.toReal_nonneg

omit [MeasureTheory.MeasureSpace M] in
/-- Localized mass on the empty set is zero. -/
@[simp] theorem massOn_empty (V : Varifold M) : massOn V ∅ = 0 := by
simp [massOn]

omit [MeasureTheory.MeasureSpace M] in
/-- Localized mass is monotone in the set. -/
theorem massOn_mono (V : Varifold M) {U W : Set M} (h : U ⊆ W) :
Expand All @@ -133,6 +138,7 @@ theorem massOn_mono (V : Varifold M) {U W : Set M} (h : U ⊆ W) :

omit [MeasureTheory.MeasureSpace M] in
/-- The total mass equals the localized mass on the whole space. -/
@[simp]
theorem massOn_univ (V : Varifold M) : massOn V Set.univ = mass V := rfl

/-- $p \in \mathrm{spt}\|V\|$ iff every open ball around $p$ has positive
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