Section ProbabilityMeasureFamily.
Context {U: Type} {SigF: SigmaAlgebraFamily U}.
Definition measurable_subspace: Type := {Omega: Ensemble U | is_measurable_subspace Omega}.
Definition measurable_subspace_Ensemble: measurable_subspace -> Ensemble U := @proj1_sig _ _.
Global Coercion measurable_subspace_Ensemble: measurable_subspace >-> Ensemble.
Definition measurable_subspace_Prop: measurable_subspace -> U -> Prop := @proj1_sig _ _.
Global Coercion measurable_subspace_Prop: measurable_subspace >-> Funclass.
Definition is_measurable_set (P: Ensemble U) (Omega: measurable_subspace) := @is_measurable_set {x: U | Omega x} (sub_sigma_algebra Omega) (sig_Set P Omega).

Lemma is_measurable_subspace_consi: forall (Omega Omega': measurable_subspace), Included Omega' Omega -> is_measurable_set Omega' Omega.