Theorems · Theorem · measure theory
MeasureTheory.integral_add_left_eq_self
∀ {G : Type u_4} {E : Type u_5} [inst : MeasurableSpace G] [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace ℝ E]
{μ : MeasureTheory.Measure G} [inst_3 : AddGroup G] [MeasurableAdd G] [μ.IsAddLeftInvariant] (f : G → E) (g : G),
∫ (x : G), f (g + x) ∂μ = ∫ (x : G), f x ∂μTranslating a function by left-addition does not change its integral with respect to a left-invariant measure.
- Defined in
- Mathlib.MeasureTheory.Group.Integral
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 256 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- MeasurableSpacestatement and proof · cited by 13,106
- NormedSpacestatement and proof · cited by 12,499
- MeasureTheory.Measurestatement and proof · cited by 10,939
- AddGroupstatement and proof · cited by 4,410
- MeasureTheory.integralstatement and proof · cited by 1,779
- MeasurableEmbeddingproof · cited by 170
- MeasureTheory.Measure.IsAddLeftInvariantstatement and proof · cited by 148
- MeasurableAddstatement and proof · cited by 78
- MeasurableEquiv.measurableEmbeddingproof · cited by 60
- MeasurableEmbedding.integral_mapproof · cited by 15
Cited by6
Results whose statement or proof uses this declaration.
- tendsto_integral_comp_smul_smul_of_integrable'proof · cited by 1
- MeasureTheory.integral_sub_left_eq_selfproof · cited by 1
- TopologicalAddGroup.IsSES.integral_pullback_invFun_applyproof · cited by 1
- MeasureTheory.integral_eq_zero_of_add_left_eq_negproof · cited by 0
- ContDiffBump.ae_convolution_tendsto_right_of_locallyIntegrableproof · cited by 0