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Fill finite unions in the null-set boolean algebra #711

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Fill finite unions in the null-set boolean algebra #711

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19 changes: 18 additions & 1 deletion Analysis/MeasureTheory/Section_1_4_1.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -94,7 +94,24 @@ def IsNull.boolean_algebra (d:ℕ) : ConcreteBooleanAlgebra (EuclideanSpace' d)
measurable := fun E => IsNull E ∨ IsNull Eᶜ
empty_mem := by sorry
compl_mem := by sorry
union_mem := by sorry
union_mem := fun E F hE hF => by
rcases hE with hE | hE <;> rcases hF with hF | hF
· refine Or.inl ?_
have hle : Lebesgue_outer_measure (E ∪ F) ≤
Lebesgue_outer_measure E + Lebesgue_outer_measure F := by
simpa [Fin.sum_univ_two] using
Lebesgue_outer_measure.finite_union_le (fun i : Fin 2 => if i = 0 then E else F)
simp [hE, hF] at hle
exact le_antisymm hle (Lebesgue_outer_measure.nonneg _)
· refine Or.inr (IsNull.subset hF ?_)
intro x hx
exact hx.2
· refine Or.inr (IsNull.subset hE ?_)
intro x hx
exact hx.1
· refine Or.inr (IsNull.subset hE ?_)
intro x hx
exact hx.1
}

def IsNull.lt_lebesgue_boolean_algebra (d:ℕ) :
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