Section RandomVariable.
Context {ora: RandomOracle} {SFo: SigmaAlgebraFamily RandomHistory}.
Definition RandomVarDomain {HBSFo: HistoryBasedSigF ora}: Type := PrFamily.measurable_subspace.
Definition RandomVarDomain_HistoryAntiChain {HBSFo: HistoryBasedSigF ora} (Omega: RandomVarDomain): HistoryAntiChain := Build_HistoryAntiChain _ (proj1_sig Omega) (measurable_subspace_legal _ (proj2_sig Omega)).
Global Coercion RandomVarDomain_HistoryAntiChain: RandomVarDomain >-> HistoryAntiChain.
Definition RandomVariable {HBSFo: HistoryBasedSigF ora} (Omega: RandomVarDomain) (A: Type) {SA: SigmaAlgebra A}: Type := PrFamily.MeasurableFunction Omega A.
Global Identity Coercion RandomVariable_MeasurableFunction: RandomVariable >-> PrFamily.MeasurableFunction.
Definition MeasurableSubset {HBSFo: HistoryBasedSigF ora} (Omega: RandomVarDomain): Type := PrFamily.measurable_set Omega.
Definition MeasurableSubset_HistoryAntiChain {HBSFo: HistoryBasedSigF ora} {Omega: RandomVarDomain} (P: MeasurableSubset Omega): HistoryAntiChain.
exists (PrFamily.measurable_set_Ensemble Omega P).
assert (Included (PrFamily.measurable_set_Ensemble Omega P) Omega).
+ unfold PrFamily.measurable_set_Ensemble.
intros ? [? ?].
auto.
+ eapply LegalHistoryAntiChain_Included; eauto.
apply measurable_subspace_legal.
destruct Omega; auto.
Defined.
Global Coercion MeasurableSubset_HistoryAntiChain: MeasurableSubset >-> HistoryAntiChain.
Context {HBSFo: HistoryBasedSigF ora}.

Lemma MeasurableSubset_in_domain: forall (Omega: RandomVarDomain) (P: MeasurableSubset Omega) h,
P h ->
Omega h.