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Prove addition preserves unsigned measurability #729

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Prove addition preserves unsigned measurability #729

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11 changes: 10 additions & 1 deletion Analysis/MeasureTheory/Section_1_3_2.lean
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Expand Up @@ -1794,7 +1794,16 @@ theorem UnsignedMeasurable.aeLimit {d:ℕ} {f: EuclideanSpace' d → EReal} (g:
theorem UnsignedMeasurable.comp_cts {d:ℕ} {f: EuclideanSpace' d → EReal} (hf: UnsignedMeasurable f) {φ: EReal → EReal} (hφ: Continuous φ) (hφnn : ∀ x ≥ 0, φ x ≥ 0) : UnsignedMeasurable (φ ∘ f) := by sorry

/-- Exercise 1.3.3(vii) -/
theorem UnsignedMeasurable.add {d:ℕ} {f g: EuclideanSpace' d → EReal} (hf: UnsignedMeasurable f) (hg: UnsignedMeasurable g) : UnsignedMeasurable (f + g) := by sorry
theorem UnsignedMeasurable.add {d:ℕ} {f g: EuclideanSpace' d → EReal} (hf: UnsignedMeasurable f) (hg: UnsignedMeasurable g) : UnsignedMeasurable (f + g) := by
obtain ⟨hfn, u, hu, huf⟩ := hf
obtain ⟨hgn, v, hv, hvg⟩ := hg
refine ⟨fun x => add_nonneg (hfn x) (hgn x),
fun n => u n + v n, fun n => (hu n).add (hv n), ?_⟩
intro x
have hfb : f x ≠ ⊥ := ne_of_gt (lt_of_lt_of_le EReal.bot_lt_zero (hfn x))
have hgb : g x ≠ ⊥ := ne_of_gt (lt_of_lt_of_le EReal.bot_lt_zero (hgn x))
exact (EReal.continuousAt_add (Or.inr hgb) (Or.inl hfb)).tendsto.comp
((huf x).prodMk_nhds (hvg x))

def UniformConvergesTo {X:Type*} (f: ℕ → X → EReal) (g: X → EReal) : Prop := ∀ ε:NNReal, ε > 0 → ∃ N:ℕ, ∀ n ≥ N, ∀ x, f n x > g x - ε ∧ f n x < g x + ε

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