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Fill Lebesgue positivity/empty and Dirac finite additivity #694

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Fill Lebesgue positivity/empty and Dirac finite additivity #694

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413235c
Fill Lebesgue positivity/empty and Dirac finite additivity.
Chessing234 Sep 7, 2026
feaf779
Remove obsolete empty-set simp lemma in Dirac measure
Chessing234 Sep 30, 2026
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19 changes: 15 additions & 4 deletions Analysis/MeasureTheory/Section_1_4_3.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -21,8 +21,8 @@ class FinitelyAdditiveMeasure {X:Type*} (B: ConcreteBooleanAlgebra X) where
noncomputable def FinitelyAdditiveMeasure.lebesgue (d:ℕ) : FinitelyAdditiveMeasure (LebesgueMeasurable.boolean_algebra d) :=
{
measure A := Lebesgue_measure A
measure_pos := by sorry
measure_empty := by sorry
measure_pos := fun A _ => Lebesgue_outer_measure.nonneg A
measure_empty := Lebesgue_measure.empty
measure_finite_additive := by sorry
}

Expand Down Expand Up @@ -60,8 +60,19 @@ noncomputable def FinitelyAdditiveMeasure.dirac {X:Type*} (x₀:X) (B: ConcreteB
split_ifs
· exact zero_le_one
· exact le_rfl
measure_empty := by simp [Set.not_mem_empty]
measure_finite_additive := by sorry
measure_empty := by simp
measure_finite_additive := by
intro E F _ _ hdisj
by_cases hxE : x₀ ∈ E
· have hxF : x₀ ∉ F := fun hxF =>
(Set.disjoint_left.mp hdisj hxE hxF)
have hxU : x₀ ∈ E ∪ F := Or.inl hxE
simp [hxE, hxF, hxU]
· by_cases hxF : x₀ ∈ F
· have hxU : x₀ ∈ E ∪ F := Or.inr hxF
simp [hxE, hxF, hxU]
· have hxU : x₀ ∉ E ∪ F := by simp [hxE, hxF]
simp [hxE, hxF, hxU]
}

/-- Example 1.4.23 (Zero measure) -/
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