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fpvandoorn/carleson
Source indexedlemma · leanprover/lean4:v4.32.0

MeasureTheory.setLIntegral_enorm_le_lintegral_rearrangement

Carleson.ToMathlib.Rearrangement · Carleson/ToMathlib/Rearrangement.lean:815 to 827

Mathematical statement

Exact Lean statement

lemma setLIntegral_enorm_le_lintegral_rearrangement {ε} [TopologicalSpace ε] [ENormedAddMonoid ε] {f : α → ε}
  (hf : AEStronglyMeasurable f μ) {X : Set α} (hX : MeasurableSet X) :
    ∫⁻ x in X, ‖f x‖ₑ ∂μ ≤
      ∫⁻ t in (Set.Iio (μ X)), rearrangement f t μ

Complete declaration

Lean source

Canonical source
Full Lean sourceLean 4
lemma setLIntegral_enorm_le_lintegral_rearrangement {ε} [TopologicalSpace ε] [ENormedAddMonoid ε] {f : α  ε}  (hf : AEStronglyMeasurable f μ) {X : Set α} (hX : MeasurableSet X) :    ∫⁻ x in X, ‖f x‖ₑ ∂μ       ∫⁻ t in (Set.Iio (μ X)), rearrangement f t μ := by  rw [ lintegral_indicator hX,  lintegral_indicator measurableSet_Iio]  have : (fun x  ‖f x‖ₑ) = enorm ∘ f := by    ext x    simp only [Function.comp_apply]  rw [this, Set.indicator_comp_of_zero enorm_zero]  simp only [Function.comp_apply]  rw [ lintegral_rearrangement (by measurability)]  gcongr with t  exact rearrangement_indicator_le