Documentation

Mathlib.MeasureTheory.Group.Measure

Measures on Groups #

We develop some properties of measures on (topological) groups

We also give analogues of all these notions in the additive world.

A measure μ on a measurable additive group is left invariant if the measure of left translations of a set are equal to the measure of the set itself.

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    A measure μ on a measurable group is left invariant if the measure of left translations of a set are equal to the measure of the set itself.

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      A measure μ on a measurable additive group is right invariant if the measure of right translations of a set are equal to the measure of the set itself.

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        A measure μ on a measurable group is right invariant if the measure of right translations of a set are equal to the measure of the set itself.

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          theorem MeasureTheory.forall_measure_preimage_add_iff {G : Type u_2} [MeasurableSpace G] [Add G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) :
          (∀ (g : G) (A : Set G), MeasurableSet Aμ ((fun h => g + h) ⁻¹' A) = μ A) MeasureTheory.Measure.IsAddLeftInvariant μ

          An alternative way to prove that μ is left invariant under addition.

          theorem MeasureTheory.forall_measure_preimage_mul_iff {G : Type u_2} [MeasurableSpace G] [Mul G] [MeasurableMul G] (μ : MeasureTheory.Measure G) :
          (∀ (g : G) (A : Set G), MeasurableSet Aμ ((fun h => g * h) ⁻¹' A) = μ A) MeasureTheory.Measure.IsMulLeftInvariant μ

          An alternative way to prove that μ is left invariant under multiplication.

          theorem MeasureTheory.forall_measure_preimage_add_right_iff {G : Type u_2} [MeasurableSpace G] [Add G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) :
          (∀ (g : G) (A : Set G), MeasurableSet Aμ ((fun h => h + g) ⁻¹' A) = μ A) MeasureTheory.Measure.IsAddRightInvariant μ

          An alternative way to prove that μ is right invariant under addition.

          theorem MeasureTheory.forall_measure_preimage_mul_right_iff {G : Type u_2} [MeasurableSpace G] [Mul G] [MeasurableMul G] (μ : MeasureTheory.Measure G) :
          (∀ (g : G) (A : Set G), MeasurableSet Aμ ((fun h => h * g) ⁻¹' A) = μ A) MeasureTheory.Measure.IsMulRightInvariant μ

          An alternative way to prove that μ is right invariant under multiplication.

          The image of a left invariant measure under a left additive action is left invariant, assuming that the action preserves addition.

          The image of a left invariant measure under a left action is left invariant, assuming that the action preserves multiplication.

          The image of a right invariant measure under a left additive action is right invariant, assuming that the action preserves addition.

          The image of a right invariant measure under a left action is right invariant, assuming that the action preserves multiplication.

          The image of a left invariant measure under right addition is left invariant.

          The image of a left invariant measure under right multiplication is left invariant.

          The image of a right invariant measure under left addition is right invariant.

          The image of a right invariant measure under left multiplication is right invariant.

          @[simp]
          theorem MeasureTheory.measure_preimage_add {G : Type u_2} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [MeasureTheory.Measure.IsAddLeftInvariant μ] (g : G) (A : Set G) :
          μ ((fun h => g + h) ⁻¹' A) = μ A

          We shorten this from measure_preimage_add_left, since left invariant is the preferred option for measures in this formalization.

          @[simp]
          theorem MeasureTheory.measure_preimage_mul {G : Type u_2} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [MeasureTheory.Measure.IsMulLeftInvariant μ] (g : G) (A : Set G) :
          μ ((fun h => g * h) ⁻¹' A) = μ A

          We shorten this from measure_preimage_mul_left, since left invariant is the preferred option for measures in this formalization.

          @[simp]
          theorem MeasureTheory.measure_preimage_add_right {G : Type u_2} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [MeasureTheory.Measure.IsAddRightInvariant μ] (g : G) (A : Set G) :
          μ ((fun h => h + g) ⁻¹' A) = μ A
          @[simp]
          theorem MeasureTheory.measure_preimage_mul_right {G : Type u_2} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [MeasureTheory.Measure.IsMulRightInvariant μ] (g : G) (A : Set G) :
          μ ((fun h => h * g) ⁻¹' A) = μ A
          theorem MeasureTheory.eventually_add_left_iff {G : Type u_2} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [MeasureTheory.Measure.IsAddLeftInvariant μ] (t : G) {p : GProp} :
          (∀ᵐ (x : G) ∂μ, p (t + x)) ∀ᵐ (x : G) ∂μ, p x
          theorem MeasureTheory.eventually_mul_left_iff {G : Type u_2} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [MeasureTheory.Measure.IsMulLeftInvariant μ] (t : G) {p : GProp} :
          (∀ᵐ (x : G) ∂μ, p (t * x)) ∀ᵐ (x : G) ∂μ, p x
          theorem MeasureTheory.eventually_add_right_iff {G : Type u_2} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [MeasureTheory.Measure.IsAddRightInvariant μ] (t : G) {p : GProp} :
          (∀ᵐ (x : G) ∂μ, p (x + t)) ∀ᵐ (x : G) ∂μ, p x
          theorem MeasureTheory.eventually_mul_right_iff {G : Type u_2} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [MeasureTheory.Measure.IsMulRightInvariant μ] (t : G) {p : GProp} :
          (∀ᵐ (x : G) ∂μ, p (x * t)) ∀ᵐ (x : G) ∂μ, p x
          theorem MeasureTheory.eventually_sub_right_iff {G : Type u_2} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [MeasureTheory.Measure.IsAddRightInvariant μ] (t : G) {p : GProp} :
          (∀ᵐ (x : G) ∂μ, p (x - t)) ∀ᵐ (x : G) ∂μ, p x
          theorem MeasureTheory.eventually_div_right_iff {G : Type u_2} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [MeasureTheory.Measure.IsMulRightInvariant μ] (t : G) {p : GProp} :
          (∀ᵐ (x : G) ∂μ, p (x / t)) ∀ᵐ (x : G) ∂μ, p x

          The measure A ↦ μ (- A), where - A is the pointwise negation of A.

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            The measure A ↦ μ (A⁻¹), where A⁻¹ is the pointwise inverse of A.

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              A measure is invariant under negation if - μ = μ. Equivalently, this means that for all measurable A we have μ (- A) = μ A, where - A is the pointwise negation of A.

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                A measure is invariant under inversion if μ⁻¹ = μ. Equivalently, this means that for all measurable A we have μ (A⁻¹) = μ A, where A⁻¹ is the pointwise inverse of A.

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                  @[simp]

                  If a left-invariant measure gives positive mass to a compact set, then it gives positive mass to any open set.

                  If a left-invariant measure gives positive mass to a compact set, then it gives positive mass to any open set.

                  abbrev MeasureTheory.isOpenPosMeasure_of_addLeftInvariant_of_regular.match_1 {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] {μ : MeasureTheory.Measure G} (motive : (K, IsCompact K μ K 0) → Prop) :
                  (x : K, IsCompact K μ K 0) → ((K : Set G) → (hK : IsCompact K) → (h2K : μ K 0) → motive (_ : K, IsCompact K μ K 0)) → motive x
                  Instances For

                    If a left-invariant measure gives finite mass to a nonempty open set, then it gives finite mass to any compact set.

                    If a left-invariant measure gives finite mass to a nonempty open set, then it gives finite mass to any compact set.

                    If a left-invariant measure gives finite mass to a set with nonempty interior, then it gives finite mass to any compact set.

                    If a left-invariant measure gives finite mass to a set with nonempty interior, then it gives finite mass to any compact set.

                    @[simp]

                    In a noncompact locally compact additive group, a left-invariant measure which is positive on open sets has infinite mass.

                    @[simp]

                    In a noncompact locally compact group, a left-invariant measure which is positive on open sets has infinite mass.

                    In an abelian additive group every left invariant measure is also right-invariant. We don't declare the converse as an instance, since that would loop type-class inference, and we use IsAddLeftInvariant as the default hypothesis in abelian groups.

                    In an abelian group every left invariant measure is also right-invariant. We don't declare the converse as an instance, since that would loop type-class inference, and we use IsMulLeftInvariant as the default hypothesis in abelian groups.

                      A measure on an additive group is an additive Haar measure if it is left-invariant, and gives finite mass to compact sets and positive mass to open sets.

                      Instances

                          A measure on a group is a Haar measure if it is left-invariant, and gives finite mass to compact sets and positive mass to open sets.

                          Instances

                            Record that an additive Haar measure on a locally compact space is locally finite. This is needed as the fact that a measure which is finite on compacts is locally finite is not registered as an instance, to avoid an instance loop.

                            See Note [lower instance priority]

                            Record that a Haar measure on a locally compact space is locally finite. This is needed as the fact that a measure which is finite on compacts is locally finite is not registered as an instance, to avoid an instance loop.

                            See Note [lower instance priority].

                            If a left-invariant measure gives positive mass to some compact set with nonempty interior, then it is an additive Haar measure.

                            If a left-invariant measure gives positive mass to some compact set with nonempty interior, then it is a Haar measure.

                            The image of an additive Haar measure under a continuous surjective proper additive group homomorphism is again an additive Haar measure. See also AddEquiv.isAddHaarMeasure_map.

                            The image of a Haar measure under a continuous surjective proper group homomorphism is again a Haar measure. See also MulEquiv.isHaarMeasure_map.

                            The image of a Haar measure under map of a left additive action is again a Haar measure

                            The image of a Haar measure under map of a left action is again a Haar measure.

                            The image of a Haar measure under right multiplication is again a Haar measure.

                            A convenience wrapper for MeasureTheory.Measure.isAddHaarMeasure_map`.

                            A Haar measure on a σ-compact space is σ-finite.

                            See Note [lower instance priority]

                            A Haar measure on a σ-compact space is σ-finite.

                            See Note [lower instance priority]

                            If the zero element of an additive group is not isolated, then an additive Haar measure on this group has no atoms.

                            This applies in particular to show that an additive Haar measure on a nontrivial finite-dimensional real vector space has no atom.

                            If the neutral element of a group is not isolated, then a Haar measure on this group has no atoms.

                            The additive version of this instance applies in particular to show that an additive Haar measure on a nontrivial finite-dimensional real vector space has no atom.