(* TODO: check whether this is necessary. *)
Definition undistinguishable_sub_domain (O1 O2: RandomVarDomain): Prop :=
  Included O1 O2 /\
  ~ is_measurable_subspace (Intersection _ O2 (Complement _ O1)).Definition RandomVar_local_equiv {O1} {O2} {A: Type} {SA: SigmaAlgebra A} (v1: RandomVariable O1 A) (v2: RandomVariable O2 A) (h1 h2: RandomHistory): Prop := forall a, v1 h1 a <-> v2 h2 a.Definition RandomVar_global_equiv {O1} {O2} {A: Type} {SA: SigmaAlgebra A} (v1: RandomVariable O1 A) (v2: RandomVariable O2 A): Prop := forall h, RandomVar_local_equiv v1 v2 h h.Definition unit_space_domain: RandomVarDomain :=
  exist is_measurable_subspace unit_space_anti_chain (max_anti_chain_measurable _ unit_anti_chain_max).Definition constant_var (Omega: RandomVarDomain) {A: Type} (v: A) {SA: SigmaAlgebra A}: RandomVariable Omega A := PrFamily.MeasurableFunction_inv (ConstantFunction v).Definition unit_space_var {A: Type} (v: A) {SA: SigmaAlgebra A}: RandomVariable unit_space_domain A := constant_var _ v.Definition RandomVarMap {Omega: RandomVarDomain} {A B: Type} {SA: SigmaAlgebra A} {SB: SigmaAlgebra B} (f: MeasurableFunction A B) (v: RandomVariable Omega A): RandomVariable Omega B := PrFamily.Compose f v.

(* Specifications: *)
Lemma RandomVarMap_sound: forall {Omega: RandomVarDomain} {A B: Type} {SA: SigmaAlgebra A} {SB: SigmaAlgebra B} (f: MeasurableFunction A B) (v: RandomVariable Omega A) h b, RandomVarMap f v h b <-> exists a, v h a /\ f a b.