Theorems · Theorem · measure theory

LipschitzWith.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},
  LipschitzWith K f →
    ∀ {d : ℝ},
      0 ≤ d →
        ∀ (s : Set X),
          (MeasureTheory.Measure.hausdorffMeasure d) (f '' s) ≤ ↑K ^ d * (MeasureTheory.Measure.hausdorffMeasure d) s

If f is a K-Lipschitz map, then it increases the Hausdorff d-measures of sets at most by the factor of K ^ d.

Defined in

Mathlib.MeasureTheory.Measure.Hausdorff

Cited by

2 results in Mathlib

Foundations

Depth 212 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

EMetricSpaceEMetricSpaceMeasurableSpaceBorelSpaceMeasurableSpaceBorelSpace