fpvandoorn/carleson
Source indexedlemma · leanprover/lean4:v4.32.0
MeasureTheory.measure_enorm_mem_eq_volume_rearrangement_mem'
Carleson.ToMathlib.Rearrangement · Carleson/ToMathlib/Rearrangement.lean:404 to 424
Mathematical statement
Exact Lean statement
lemma measure_enorm_mem_eq_volume_rearrangement_mem' {ε} [TopologicalSpace ε] [ENormedAddMonoid ε]
{f : α → ε} (hf : AEStronglyMeasurable f μ) {s : Set ℝ≥0∞}
(hs : MeasurableSet s) (zero_notin_s : 0 ∉ s) (h : ∃ T, (∀ t ∈ s, T < t) ∧ distribution f T μ < ∞) :
μ {x | ‖f x‖ₑ ∈ s} = volume {x | rearrangement f x μ ∈ s}Complete declaration
Lean source
Full Lean sourceLean 4
lemma measure_enorm_mem_eq_volume_rearrangement_mem' {ε} [TopologicalSpace ε] [ENormedAddMonoid ε] {f : α → ε} (hf : AEStronglyMeasurable f μ) {s : Set ℝ≥0∞} (hs : MeasurableSet s) (zero_notin_s : 0 ∉ s) (h : ∃ T, (∀ t ∈ s, T < t) ∧ distribution f T μ < ∞) : μ {x | ‖f x‖ₑ ∈ s} = volume {x | rearrangement f x μ ∈ s} := by rcases h with ⟨T, hT, hT'⟩ have hμ : {x | ‖f x‖ₑ ∈ s} = {x | ‖(superlevelSet f T).indicator f x‖ₑ ∈ s} := by ext x unfold superlevelSet Set.indicator aesop have hvolume : {x | rearrangement f x μ ∈ s} = {x | (superlevelSet (rearrangement f · μ) T).indicator (rearrangement f · μ) x ∈ s} := by ext x unfold superlevelSet Set.indicator aesop rw [hμ, hvolume] rw [measure_enorm_mem_eq_volume_rearrangement_mem_of_support_finite (μ := μ) _ _ hs zero_notin_s] · congr with x rw [rearrangement_indicator_superlevelSet] · exact hf.indicator₀ (nullMeasurableSet_superlevelSet hf) · rw [← distribution_zero_eq_measure_support, distribution_indicator_superlevelSet] exact min_lt_of_left_lt hT'