Theorems · Theorem · measure theory

Real.smul_map_diagonal_volume_pi

∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {D : ι → ℝ},
  (Matrix.diagonal D).det ≠ 0 →
    ENNReal.ofReal |(Matrix.diagonal D).det| •
        MeasureTheory.Measure.map (⇑(Matrix.toLin' (Matrix.diagonal D))) MeasureTheory.volume =
      MeasureTheory.volume

A diagonal matrix rescales Lebesgue according to its determinant. This is a special case of Real.map_matrix_volume_pi_eq_smul_volume_pi, that one should use instead (and whose proof uses this particular case).

Defined in: Mathlib.MeasureTheory.Measure.Lebesgue.Basic
Cited by: 1 results in Mathlib
Foundations: Depth 250 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Fintype DecidableEq

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites48

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Setproof · cited by 53,352
Realstatement and proof · cited by 25,697
RingHom.idstatement · cited by 18,349
MeasureTheory.Measurestatement · cited by 10,939
LinearMapstatement · cited by 10,215
ENNRealstatement and proof · cited by 9,879
Fintypestatement and proof · cited by 7,736
Set.preimageproof · cited by 4,946
Matrixstatement · cited by 4,303
Set.univproof · cited by 3,945
Finset.univproof · cited by 3,473

Cited by1

Results whose statement or proof uses this declaration.

Real.map_matrix_volume_pi_eq_smul_volume_piproof · cited by 1