{"name":"mem_symmGroup","declaration":"theorem mem_symmGroup {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [AddCommGroup G] [MeasurableSpace G] [MeasurableAdd₂ G] {X : Ω → G} (hX : Measurable X) {x : G} : x ∈ symmGroup X hX ↔ ProbabilityTheory.IdentDistrib X (fun ω => X ω + x) MeasureTheory.volume MeasureTheory.volume"}
{"name":"exists_isUniform_of_rdist_eq_zero","declaration":"/-- If $d[X_1;X_2]=0$, then there exists a subgroup $H \\leq G$ such that\n$d[X_1;U_H] = d[X_2;U_H] = 0$. Follows from the preceding claim by the triangle inequality. -/\ntheorem exists_isUniform_of_rdist_eq_zero {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableAdd₂ G] [MeasurableSub₂ G] {X : Ω → G} [MeasurableSingletonClass G] {Ω' : Type u_3} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X' : Ω' → G} (hX : Measurable X) (hX' : Measurable X') (hdist : d[X # X'] = 0) : ∃ H U, Measurable U ∧ ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧ d[X # U] = 0 ∧ d[X' # U] = 0"}
{"name":"sub_mem_symmGroup","declaration":"/-- If $d[X ;X]=0$, and $x,y \\in G$ are such that $P[X=x], P[X=y]>0$,\nthen $x-y \\in \\mathrm{Sym}[X]$. -/\ntheorem sub_mem_symmGroup {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableAdd₂ G] [MeasurableSub₂ G] {X : Ω → G} [MeasurableSingletonClass G] (hX : Measurable X) (hdist : d[X # X] = 0) {x : G} {y : G} (hx : ↑↑MeasureTheory.volume (X ⁻¹' {x}) ≠ 0) (hy : ↑↑MeasureTheory.volume (X ⁻¹' {y}) ≠ 0) : x - y ∈ symmGroup X hX"}
{"name":"exists_isUniform_of_rdist_self_eq_zero","declaration":"/-- If $d[X ;X]=0$, then there exists a subgroup $H \\leq G$ such that $d[X ;U_H] = 0$. -/\ntheorem exists_isUniform_of_rdist_self_eq_zero {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableAdd₂ G] [MeasurableSub₂ G] {X : Ω → G} [MeasurableSingletonClass G] (hX : Measurable X) (hdist : d[X # X] = 0) : ∃ H U, Measurable U ∧ ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧ d[X # U] = 0"}
{"name":"symmGroup","declaration":"/-- The symmetry group Sym of $X$: the set of all $h ∈ G$ such that $X + h$ has an identical\ndistribution to $X$. -/\ndef symmGroup {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [AddCommGroup G] [MeasurableSpace G] [MeasurableAdd₂ G] (X : Ω → G) (hX : Measurable X) : AddSubgroup G"}
{"name":"ProbabilityTheory.IdentDistrib.symmGroup_eq","declaration":"theorem ProbabilityTheory.IdentDistrib.symmGroup_eq {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [AddCommGroup G] [MeasurableSpace G] [MeasurableAdd₂ G] {X : Ω → G} {Ω' : Type u_3} [MeasureTheory.MeasureSpace Ω'] {X' : Ω' → G} (hX : Measurable X) (hX' : Measurable X') (h : ProbabilityTheory.IdentDistrib X X' MeasureTheory.volume MeasureTheory.volume) : symmGroup X hX = symmGroup X' hX'"}
{"name":"isUniform_sub_const_of_rdist_eq_zero","declaration":"/-- If `d[X # X] = 0`, then `X - x₀` is the uniform distribution on the subgroup of `G`\nstabilizing the distribution of `X`, for any `x₀` of positive probability. -/\ntheorem isUniform_sub_const_of_rdist_eq_zero {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableAdd₂ G] [MeasurableSub₂ G] {X : Ω → G} [MeasurableSingletonClass G] (hX : Measurable X) (hdist : d[X # X] = 0) {x₀ : G} (hx₀ : ↑↑MeasureTheory.volume (X ⁻¹' {x₀}) ≠ 0) : ProbabilityTheory.IsUniform (↑(symmGroup X hX)) (fun ω => X ω - x₀) MeasureTheory.volume"}