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Fill Lebesgue outer measure of the empty set and monotonicity #690

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Fill Lebesgue outer measure of the empty set and monotonicity #690

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9 changes: 7 additions & 2 deletions Analysis/MeasureTheory/Section_1_2_1.lean
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Expand Up @@ -50,12 +50,17 @@ lemma Box.isClosed_toSet_of_Icc {d : ℕ} (B : Box d)

/-- Exercise 1.2.3(i) (Empty set) -/
theorem Lebesgue_outer_measure.of_empty (d:ℕ) : Lebesgue_outer_measure (∅: Set (EuclideanSpace' d)) = 0 := by
sorry
obtain rfl | hd := Nat.eq_zero_or_pos d
· simp [Lebesgue_outer_measure_of_dim_zero]
· exact Countable.Lebesgue_measure hd Set.countable_empty

/-- Exercise 1.2.3(ii) (Monotonicity) -/
theorem Lebesgue_outer_measure.mono {d: ℕ} {E F : Set (EuclideanSpace' d)} (h : E ⊆ F) :
Lebesgue_outer_measure E ≤ Lebesgue_outer_measure F := by
sorry
unfold Lebesgue_outer_measure
refine sInf_le_sInf ?_
rintro V ⟨X, S, hF, rfl⟩
exact ⟨X, S, h.trans hF, rfl⟩

/-- Lebesgue outer measure is non-negative.
Since it's the sInf of sums of box volumes, which are all ≥ 0, the result is ≥ 0. -/
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