Theorems · Theorem · measure theory
LinearIsometryEquiv.measurePreserving
∀ {E : Type u_2} {F : Type u_3} [inst : NormedAddCommGroup F] [inst_1 : InnerProductSpace ℝ F]
[inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E]
[inst_6 : MeasurableSpace F] [inst_7 : BorelSpace F] [inst_8 : FiniteDimensional ℝ E] [inst_9 : FiniteDimensional ℝ F]
(f : E ≃ₗᵢ[ℝ] F), MeasureTheory.MeasurePreserving (⇑f) MeasureTheory.volume MeasureTheory.volumeEvery linear isometry on a real finite-dimensional Hilbert space is measure-preserving.
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 265 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites29
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
- Realstatement and proof · cited by 25,697
- RingHom.idstatement and proof · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- Finsetproof · cited by 13,712
- MeasurableSpacestatement and proof · cited by 13,106
- NormedSpaceproof · cited by 12,499
- MeasureTheory.Measureproof · cited by 10,939
- Fintypeproof · cited by 7,736
- InnerProductSpacestatement and proof · cited by 3,523
- FiniteDimensionalstatement and proof · cited by 1,854
- BorelSpacestatement and proof · cited by 1,602
Cited by8
Results whose statement or proof uses this declaration.
- Real.fourier_comp_linearIsometryproof · cited by 2
- GaussianFourier.integral_cexp_neg_mul_sq_norm_addproof · cited by 2
- WithLp.volume_preserving_symm_measurableEquiv_toLp_prodproof · cited by 2
- MeasureTheory.volume_eq_of_finrank_eq_oneproof · cited by 1
- GaussianFourier.integrable_cexp_neg_mul_sq_norm_addproof · cited by 1
- Submodule.measurePreserving_measurableEquivProdproof · cited by 1
- MeasureTheory.integral_compproof · cited by 0
- MeasureTheory.integrable_compproof · cited by 0