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feat(analysis/convex): add dual cone (#7738) Add definition of the dual cone of a set in a real inner product space
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feat(analysis/normed_space): nnreal.coe_nat_abs (#7911) from LTE
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feat(analysis/convex): add dual cone (#7738) Add definition of the dual cone of a set in a real inner product space
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Add definition of the dual cone of a set in a real inner product space
Hello! I am not sure if this is the "right" definition of a dual cone. An alternative would be to define the dual cone as a cone in the dual vector space and to use y(x) instead of ⟪ x, y ⟫.
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feat(analysis/convex): add dual cone
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This PR changes slightly the doc-strings to make the autogenerated documentation more consistent. I also removed an unstylish double space.
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feat(data/setoid/partition): indexed partition
_ (subtype (@is_partition α)) _ (partial_order.to_preorder _) $ partition.rel_i end partition end setoid++/-- A partition of a type indexed by another type. -/+@[nolint has_inhabited_instance]+structure indexed_partition {ι α : Type*} (s : ι → set α) :=+(eq_of_mem : ∀ {x i j}, x ∈ s i → x ∈ s j → i = j)+(some : ι → α)+(some_mem : ∀ i, some i ∈ s i)+(index : α → ι)+(mem_index : ∀ x, x ∈ s (index x))++/-- The non-constructive constructor for `indexed_partition`. -/+noncomputable+def indexed_partition.mk' {ι α : Type*} (s : ι → set α) (dis : ∀ i j, i ≠ j → disjoint (s i) (s j))+(empty : ∀ i, (s i).nonempty) (ex : ∀ x, ∃ i, x ∈ s i) : indexed_partition s :=+{ eq_of_mem := begin+ classical,+ intros x i j hxi hxj,+ by_contra h,+ exact dis _ _ h ⟨hxi, hxj⟩+ end,+ some := λ i, (empty i).some,+ some_mem := λ i, (empty i).some_spec,+ index := λ x, (ex x).some,+ mem_index := λ x, (ex x).some_spec }++namespace indexed_partition++open set++variables {ι α : Type*} {s : ι → set α} (hs : indexed_partition s)+include hs++lemma exists_mem (x : α) : ∃ i, x ∈ s i := ⟨hs.index x, hs.mem_index x⟩++lemma Union : (⋃ i, s i) = univ :=+by { ext x, simp [hs.exists_mem x] }++lemma disjoint : ∀ {i j}, i ≠ j → disjoint (s i) (s j) :=+λ i j h x ⟨hxi, hxj⟩,h (hs.eq_of_mem hxi hxj)
λ i j h x ⟨hxi, hxj⟩, h (hs.eq_of_mem hxi hxj)
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feat(measure_theory): define `measure.trim`, restriction of a measure to a sub-sigma algebra
attribute [integral_simps] integral_neg integral_smul L1.integral_add L1.integra attribute [irreducible] integral L1.integral +section integral_trim++variables {H β γ : Type*} [normed_group H] [measurable_space H]+ {m m0 : measurable_space β} {μ : measure β}++/-- Simple function seen as simple function of a larger `measurable_space`. -/+def simple_func.to_larger_space (hm : m ≤ m0) (f : @simple_func β m γ) : simple_func β γ :=+⟨@simple_func.to_fun β m γ f, λ x, hm _ (@simple_func.measurable_set_fiber β γ m f x),+ @simple_func.finite_range β γ m f⟩++lemma simple_func.coe_to_larger_space_eq (hm : m ≤ m0) (f : @simple_func β m γ) :+ ⇑(f.to_larger_space hm) = f :=+rfl++lemma integral_simple_func_larger_space (hm : m ≤ m0) (f : @simple_func β m F)+ (hf_int : integrable f μ) :+ ∫ x, f x ∂μ = ∑ x in (@simple_func.range β F m f), (ennreal.to_real (μ (f ⁻¹' {x}))) • x :=+begin+ simp_rw ← f.coe_to_larger_space_eq hm,+ have hf_int : integrable (f.to_larger_space hm) μ, by rwa simple_func.coe_to_larger_space_eq,+ rw simple_func.integral_eq_sum _ hf_int,+ congr,+end++lemma integral_trim_simple_func (hm : m ≤ m0) (f : @simple_func β m F) (hf_int : integrable f μ) :+ ∫ x, f x ∂μ = @integral β F m _ _ _ _ _ _ (μ.trim hm) f :=+begin+ have hf : @measurable _ _ m _ f, from @simple_func.measurable β F m _ f,+ have hf_int_m := hf_int.trim hm hf,+ rw [integral_simple_func_larger_space le_rfl f hf_int_m,+ integral_simple_func_larger_space hm f hf_int],+ congr,+ ext1 x,+ congr,+ exact (trim_measurable_set_eq hm (@simple_func.measurable_set_fiber β F m f x)).symm,+end++lemma integral_trim (hm : m ≤ m0) {f : β → F} (hf : @measurable β F m _ f)+ (hf_int : integrable f μ) :
Thanks for spotting this! Indeed I could remove the integrability assumption, here and in the two other results.
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feat(measure_theory): define `measure.trim`, restriction of a measure to a sub-sigma algebra
attribute [integral_simps] integral_neg integral_smul L1.integral_add L1.integra attribute [irreducible] integral L1.integral +section integral_trim++variables {H β γ : Type*} [normed_group H] [measurable_space H]+ {m m0 : measurable_space β} {μ : measure β}++/-- Simple function seen as simple function of a larger `measurable_space`. -/+def simple_func.to_larger_space (hm : m ≤ m0) (f : @simple_func β m γ) : simple_func β γ :=+⟨@simple_func.to_fun β m γ f, λ x, hm _ (@simple_func.measurable_set_fiber β γ m f x),+ @simple_func.finite_range β γ m f⟩++lemma simple_func.coe_to_larger_space_eq (hm : m ≤ m0) (f : @simple_func β m γ) :+ ⇑(f.to_larger_space hm) = f :=+rfl++lemma integral_simple_func_larger_space (hm : m ≤ m0) (f : @simple_func β m F)+ (hf_int : integrable f μ) :+ ∫ x, f x ∂μ = ∑ x in (@simple_func.range β F m f), (ennreal.to_real (μ (f ⁻¹' {x}))) • x :=+begin+ simp_rw ← f.coe_to_larger_space_eq hm,+ have hf_int : integrable (f.to_larger_space hm) μ, by rwa simple_func.coe_to_larger_space_eq,+ rw simple_func.integral_eq_sum _ hf_int,+ congr,+end++lemma integral_trim_simple_func (hm : m ≤ m0) (f : @simple_func β m F) (hf_int : integrable f μ) :+ ∫ x, f x ∂μ = @integral β F m _ _ _ _ _ _ (μ.trim hm) f :=+begin+ have hf : @measurable _ _ m _ f, from @simple_func.measurable β F m _ f,+ have hf_int_m := hf_int.trim hm hf,+ rw [integral_simple_func_larger_space le_rfl f hf_int_m,+ integral_simple_func_larger_space hm f hf_int],+ congr,+ ext1 x,+ congr,+ exact (trim_measurable_set_eq hm (@simple_func.measurable_set_fiber β F m f x)).symm,+end++lemma integral_trim (hm : m ≤ m0) {f : β → F} (hf : @measurable β F m _ f)+ (hf_int : integrable f μ) :
Is the integrability assumption really necessary here? If f is not integrable with respect to the big sigma-algebra, it won't either with respect to the small sigma-algebra, and both integrals will vanish, right?
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Pull request review commentleanprover-community/mathlib
feat(measure_theory): define `measure.trim`, restriction of a measure to a sub-sigma algebra
end end is_complete +namespace measure_theory++lemma outer_measure.to_measure_zero [measurable_space α] : (0 : outer_measure α).to_measure+ ((le_top).trans outer_measure.zero_caratheodory.symm.le) = 0 :=+by rw [← measure.measure_univ_eq_zero, to_measure_apply _ _ measurable_set.univ,+ outer_measure.coe_zero, pi.zero_apply]++section trim++/-- Restriction of a measure to a sub-sigma algebra.+It is common to see a measure `μ` on a measurable space structure `m0` as being also a measure on+any `m ≤ m0`. Since measures in mathlib have to be trimmed to the measurable space, `μ` itself+cannot be a measure on `m`, hence the definition of `μ.trim hm`.++This notion is related to `outer_measure.trim`, see the lemma+`to_outer_measure_trim_eq_trim_to_outer_measure`. -/+def measure.trim {m m0 : measurable_space α} (μ : @measure α m0) (hm : m ≤ m0) : @measure α m :=+@outer_measure.to_measure α m μ.to_outer_measure (hm.trans (le_to_outer_measure_caratheodory μ))++@[simp] lemma trim_eq_self [measurable_space α] {μ : measure α} : μ.trim le_rfl = μ :=+by simp [measure.trim]++variables {m m0 : measurable_space α} {μ : measure α} {s : set α}++lemma to_outer_measure_trim_eq_trim_to_outer_measure (μ : measure α) (hm : m ≤ m0) :+ @measure.to_outer_measure _ m (μ.trim hm) = @outer_measure.trim _ m μ.to_outer_measure :=+by rw [measure.trim, to_measure_to_outer_measure]++@[simp] lemma zero_trim (hm : m ≤ m0) : (0 : measure α).trim hm = (0 : @measure α m) :=+by simp [measure.trim, outer_measure.to_measure_zero]++lemma trim_measurable_set_eq (hm : m ≤ m0) (hs : @measurable_set α m s) : μ.trim hm s = μ s :=+by simp [measure.trim, hs]++lemma le_trim (hm : m ≤ m0) : μ s ≤ μ.trim hm s :=+by {simp_rw [measure.trim], exact (@le_to_measure_apply _ m _ _ _), }
by { simp_rw [measure.trim], exact (@le_to_measure_apply _ m _ _ _), }
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feat(measure_theory): define `measure.trim`, restriction of a measure to a sub-sigma algebra
begin exact hfi.ae_measurable.null_measurable_set (measurable_set_singleton 0).compl end +lemma set_integral_trim {α} {m m0 : measurable_space α} {μ : measure α} (hm : m ≤ m0) {f : α → E}
same comment as above: I am not sure that you need the integrability assumption.
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feat(measure_theory): define `measure.trim`, restriction of a measure to a sub-sigma algebra
attribute [integral_simps] integral_neg integral_smul L1.integral_add L1.integra attribute [irreducible] integral L1.integral +section integral_trim++variables {H β γ : Type*} [normed_group H] [measurable_space H]+ {m m0 : measurable_space β} {μ : measure β}++/-- Simple function seen as simple function of a larger `measurable_space`. -/+def simple_func.to_larger_space (hm : m ≤ m0) (f : @simple_func β m γ) : simple_func β γ :=+⟨@simple_func.to_fun β m γ f, λ x, hm _ (@simple_func.measurable_set_fiber β γ m f x),+ @simple_func.finite_range β γ m f⟩++lemma simple_func.coe_to_larger_space_eq (hm : m ≤ m0) (f : @simple_func β m γ) :+ ⇑(f.to_larger_space hm) = f :=+rfl++lemma integral_simple_func_larger_space (hm : m ≤ m0) (f : @simple_func β m F)+ (hf_int : integrable f μ) :+ ∫ x, f x ∂μ = ∑ x in (@simple_func.range β F m f), (ennreal.to_real (μ (f ⁻¹' {x}))) • x :=+begin+ simp_rw ← f.coe_to_larger_space_eq hm,+ have hf_int : integrable (f.to_larger_space hm) μ, by rwa simple_func.coe_to_larger_space_eq,+ rw simple_func.integral_eq_sum _ hf_int,+ congr,+end++lemma integral_trim_simple_func (hm : m ≤ m0) (f : @simple_func β m F) (hf_int : integrable f μ) :+ ∫ x, f x ∂μ = @integral β F m _ _ _ _ _ _ (μ.trim hm) f :=+begin+ have hf : @measurable _ _ m _ f, from @simple_func.measurable β F m _ f,+ have hf_int_m := hf_int.trim hm hf,+ rw [integral_simple_func_larger_space le_rfl f hf_int_m,+ integral_simple_func_larger_space hm f hf_int],+ congr,+ ext1 x,+ congr,+ exact (trim_measurable_set_eq hm (@simple_func.measurable_set_fiber β F m f x)).symm,+end++lemma integral_trim (hm : m ≤ m0) {f : β → F} (hf : @measurable β F m _ f)+ (hf_int : integrable f μ) :+ ∫ x, f x ∂μ = @integral β F m _ _ _ _ _ _ (μ.trim hm) f :=+begin+ let f_seq := @simple_func.approx_on F β _ _ _ m _ hf set.univ 0 (set.mem_univ 0) _,+ have hf_seq_meas : ∀ n, @measurable _ _ m _ (f_seq n),+ from λ n, @simple_func.measurable β F m _ (f_seq n),+ have hf_seq_int : ∀ n, integrable (f_seq n) μ,+ from simple_func.integrable_approx_on_univ (hf.mono hm le_rfl) hf_int,+ have hf_seq_int_m : ∀ n, @integrable β F m _ _ (f_seq n) (μ.trim hm),+ from λ n, (hf_seq_int n).trim hm (hf_seq_meas n) ,+ have hf_seq_eq : ∀ n, ∫ x, f_seq n x ∂μ = @integral β F m _ _ _ _ _ _ (μ.trim hm) (f_seq n),+ from λ n, integral_trim_simple_func hm (f_seq n) (hf_seq_int n),+ have h_lim_1 : at_top.tendsto (λ n, ∫ x, f_seq n x ∂μ) (𝓝 (∫ x, f x ∂μ)),+ { refine tendsto_integral_of_L1 f hf_int (eventually_of_forall hf_seq_int) _,+ exact simple_func.tendsto_approx_on_univ_L1_edist (hf.mono hm le_rfl) hf_int, },+ have h_lim_2 : at_top.tendsto (λ n, ∫ x, f_seq n x ∂μ)+ (𝓝 (@integral β F m _ _ _ _ _ _ (μ.trim hm) f)),+ { simp_rw hf_seq_eq,+ refine @tendsto_integral_of_L1 β F m _ _ _ _ _ _ (μ.trim hm) _ f+ (hf_int.trim hm hf) _ _ (eventually_of_forall hf_seq_int_m) _,+ exact @simple_func.tendsto_approx_on_univ_L1_edist β F m _ _ _ _ f _ hf (hf_int.trim hm hf), },+ exact tendsto_nhds_unique h_lim_1 h_lim_2,+end++lemma integral_trim' (hm : m ≤ m0) {f : β → F} (hf : @integrable β F m _ _ f (μ.trim hm)) :
If my comment above is true, ae_measurability should be enough for this one.
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feat(geometry/manifold): Some lemmas for smooth functions (#7752)
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feat(linear_algebra/free_module): add class module.free (#7801) We introduce here a new class `module.free`. Co-authored-by: Johan Commelin <[email protected]>
commit sha ee4fe7432b1f88e857e4b568f01a6435276a1b96
feat(topology/category/Profinite/cofiltered_clopen): Theorem about clopen sets in cofiltered limits of profinite sets (#7837) This PR proves the theorem that any clopen set in a cofiltered limit of profinite sets arises from a clopen set in one of the factors of the limit. This generalizes a theorem used in LTE.
commit sha 39073fa25e35ba2edade7b5f2aca6ffce944ae89
feat(algebra/pointwise): Dynamics of powers of a subset (#7836) If `S` is a subset of a group `G`, then the powers of `S` eventually stabilize in size. Co-authored-by: tb65536 <[email protected]>
commit sha 7b7cd0a140c51844aa4d5a7d6ea15cb5f6e1afd7
fix(tactic/lint): punctuation of messages (#7869) Previously, the linter framework would append punctuation (`.` or `:`) to the message provided by the linter, but this was confusing and lead to some double punctuation. Now all linters specify their own punctuation. Co-authored-by: Floris van Doorn <[email protected]>
commit sha 9ad8ea3b6275535ded9b7bfd315300a031ed326a
chore(linear_algebra/quadratic_form): fix typo (#7907)
commit sha e0a3303f6c7530dcebf38577085b02e9c490bbd4
chore(category_theory/filtered): Adds missing instances (#7909) Co-authored-by: Patrick Massot <[email protected]>
commit sha add577d77d36a132e10f4b0903bddd8db0e07ac6
feat(group_theory/group_action/defs): add `has_mul.to_has_scalar` and relax typeclass in `smul_mul_smul` (#7885)
commit sha 2c919b0874453791978075e76dbdee23a56f0201
chore(algebra/{ordered_group, linear_ordered_comm_group_with_zero.lean}): rename one lemma, remove more @s (#7895) The more substantial part of this PR is changing the name of a lemma from `div_lt_div_iff'` to `mul_inv_lt_mul_inv_iff'`: the lemma proves `a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b`. Furthermore, in the same spirit as a couple of my recent short PRs, I am removing a few more `@`, in order to sweep under the rug, later on, a change in typeclass assumptions. This PR only changes a name, which was used only once, and a few proofs, but no statement. On the path towards PR #7645.
commit sha e13fd486195fbc7a949161c20c2449a3216d1e1e
docs(data/nat/pairing): add module docstring (#7897)
commit sha 7c9643def9729216fd46352606e1584c2d991aca
chore(scripts): update nolints.txt (#7914) I am happy to remove some nolints for you!
commit sha 6d2a0512c4bcd898cc179c1b4603c5a00fa3e285
feat(algebra/covariant_and_contravariant): API for covariant_and_contravariant (#7889) This PR introduces more API for `covariant` and `contravariant` stuff . Besides the API, I have not actually made further use of the typeclasses or of the API. This happens in subsequent PRs. This is a step towards PR #7645.
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feat(measure_theory): continuity of primitives (#7864) From the sphere eversion project This proves some continuity of interval integrals with respect to parameters and continuity of primitives of measurable functions. The statements are a bit abstract, but they allow to have: ```lean example {f : ℝ → E} (h_int : integrable f) (a : ℝ) : continuous (λ b, ∫ x in a .. b, f x ∂ volume) := h_int.continuous_primitive a ``` under the usual assumptions on `E`: `normed_group E`, `second_countable_topology E`, `normed_space ℝ E` `complete_space E`, `measurable_space E`, `borel_space E`, say `E = ℝ` for instance. Of course global integrability is not needed, assuming integrability on all finite length intervals is enough: ```lean example {f : ℝ → E} (h_int : ∀ a b : ℝ, interval_integrable f volume a b) (a : ℝ) : continuous (λ b, ∫ x in a .. b, f x ∂ volume) := continuous_primitive h_int a ```
commit sha 5c114586544998ea34d18cfe63700b5fe7507eb4
chore(analysis/normed_space/normed_group_hom): golf proof of normed_group_hom.bounded (#7896)
commit sha a359bd989d3bc0917c282b59ba86dfafd5f26c4a
chore(measure_theory): measurability statements for coercions, coherent naming (#7854) Also add a few lemmas on measure theory
commit sha e971eae617baf2c358b32fe3dff33798e00e446f
docs(data/nat/totient): add module docstring (#7899)
commit sha b3244884a52c0dfa2ca48825fb45bcf85f9551e3
docs(set_theory/schroeder_bernstein): add module docstring (#7900)
commit sha 4fe77812bd05841b05b7dd5ac43cb5faf4917255
chore(algebra/lie/basic + classical): golf some proofs (#7903) Another PR with some golfing, to get acquainted with the files! Oliver, I really like how you set this up! Also, feel free to say that you do not like the golfing: there is a subtle tension between proving stuff fast and making it accessible!
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feat(measure_theory/ess_sup): monotonicity of ess_sup/ess_inf w.r.t. the measure (#7917)
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chore(scripts): update nolints.txt (#7922) I am happy to remove some nolints for you!
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feat(analysis/normed_space): nnreal.coe_nat_abs
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docs(data/rat/denumerable): add module docstring
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