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 μComplete declaration
Lean 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