proposition: Lemma le_measure d (R : realFieldType) (T : semiRingOfSetsType d) (mu : {content set T -> \bar R}) : {in measurable &, {homo mu : A B / A `<=` B >-> (A <= B)%E}}.
proof: Proof. move=> A B; rewrite ?inE => mA mB AB; have [|muBfin] := leP +oo%E (mu B). by rewrite leye_eq => /eqP ->; rewrite leey. rewrite -[leRHS]SetRing.RmuE// -[B](setDUK AB) measureU/= ?setDIK//. - by rewrite SetRing.RmuE ?leeDl. - exact: sub_gen_smallest. - by apply: measurableD; exact: sub_gen_smallest. Qed.
imports: mathcomp(all_ssreflect all_algebra finmap) mathcomp(mathcomp_extra boolp classical_sets functions) mathcomp(cardinality fsbigop ) reals ereal signed topology normedtype sequences esum numfun HB(structures)