Theorems · Theorem · measure theory

LipschitzOnWith.hausdorffMeasure_image_le

∀ {X : Type u_2} {Y : Type u_3} [inst : EMetricSpace X] [inst_1 : EMetricSpace Y] [inst_2 : MeasurableSpace X]
  [inst_3 : BorelSpace X] [inst_4 : MeasurableSpace Y] [inst_5 : BorelSpace Y] {K : NNReal} {f : X → Y} {s : Set X},
  LipschitzOnWith K f s →
    ∀ {d : ℝ},
      0 ≤ d →
        (MeasureTheory.Measure.hausdorffMeasure d) (f '' s) ≤ ↑K ^ d * (MeasureTheory.Measure.hausdorffMeasure d) s

If f : X → Y is K-Lipschitz on s, then μH[d] (f '' s) ≤ K ^ d * μH[d] s.

Defined in

Mathlib.MeasureTheory.Measure.Hausdorff

Cited by

1 results in Mathlib

Foundations

Depth 211 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

EMetricSpaceEMetricSpaceMeasurableSpaceBorelSpaceMeasurableSpaceBorelSpace