Theorems · Theorem · measure theory

HolderOnWith.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] {C r : NNReal} {f : X → Y} {s : Set X},
  HolderOnWith C r f s →
    0 < r →
      ∀ {d : ℝ},
        0 ≤ d →
          (MeasureTheory.Measure.hausdorffMeasure d) (f '' s) ≤
            ↑C ^ d * (MeasureTheory.Measure.hausdorffMeasure (↑r * d)) s

If f : X → Y is Hölder continuous on s with a positive exponent r, then μH[d] (f '' s) ≤ C ^ d * μH[r * d] s.

Defined in

Mathlib.MeasureTheory.Measure.Hausdorff

Cited by

2 results in Mathlib

Foundations

Depth 210 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

EMetricSpaceEMetricSpaceMeasurableSpaceBorelSpaceMeasurableSpaceBorelSpace