Theorems · Theorem · functional analysis

LinearMap.hausdorffMeasure_image

∀ {U : Type u_5} {V : Type u_6} [inst : NormedAddCommGroup U] [inst_1 : InnerProductSpace ℝ U]
  [inst_2 : FiniteDimensional ℝ U] [inst_3 : NormedAddCommGroup V] [inst_4 : InnerProductSpace ℝ V]
  [inst_5 : MeasurableSpace U] [inst_6 : BorelSpace U] [inst_7 : MeasurableSpace V] [inst_8 : BorelSpace V]
  (f : U →ₗ[ℝ] V) (s : Set U),
  (MeasureTheory.Measure.hausdorffMeasure ↑(Module.finrank ℝ U)) (⇑f '' s) =
    ENNReal.ofReal f.normDet * (MeasureTheory.Measure.hausdorffMeasure ↑(Module.finrank ℝ U)) s

Using Hausdorff measure with the domain dimension, the volume of the image is scaled by LinearMap.normDet.

Defined in

Mathlib.Analysis.InnerProductSpace.NormDet

Cited by

1 results in Mathlib

Foundations

Depth 259 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceFiniteDimensionalNormedAddCommGroupInnerProductSpaceMeasurableSpaceBorelSpaceMeasurableSpaceBorelSpace