diff --git a/theories/kernel.v b/theories/kernel.v index 5ae016504e..9a72194fe6 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -846,16 +846,14 @@ Context d1 d2 d3 (X : measurableType d1) (Y : measurableType d2) Variable l : R.-ker X ~> Y. Variable k : R.-ker [the measurableType _ of X * Y] ~> Z. -Local Notation "l \; k" := (kcomp l k). - -Let kcomp0 x : (l \; k) x set0 = 0. +Let kcomp0 x : kcomp l k x set0 = 0. Proof. by rewrite /kcomp (eq_integral (cst 0)) ?integral0// => y _; rewrite measure0. Qed. -Let kcomp_ge0 x U : 0 <= (l \; k) x U. Proof. exact: integral_ge0. Qed. +Let kcomp_ge0 x U : 0 <= kcomp l k x U. Proof. exact: integral_ge0. Qed. -Let kcomp_sigma_additive x : semi_sigma_additive ((l \; k) x). +Let kcomp_sigma_additive x : semi_sigma_additive (kcomp l k x). Proof. move=> U mU tU mUU; rewrite [X in _ --> X](_ : _ = \int[l x]_y (\sum_(n {measure set Z -> \bar R} := fun x => - [the measure _ _ of (l \; k) x]. + [the measure _ _ of kcomp l k x]. End kcomp_is_measure. @@ -902,7 +900,7 @@ Context d d' d3 (X : measurableType d) (Y : measurableType d') Variable l : R.-fker X ~> Y. Variable k : R.-fker [the measurableType _ of X * Y] ~> Z. -Let mkcomp_finite : measure_fam_uub (l \; k). +Let mkcomp_finite : measure_fam_uub (kcomp l k). Proof. have /measure_fam_uubP[r hr] := measure_uub k. have /measure_fam_uubP[s hs] := measure_uub l. @@ -1047,14 +1045,14 @@ Variables (l : R.-sfker X ~> Y) (k : R.-sfker [the measurableType _ of X * Y] ~> Z). Let integral_kcomp_indic x E (mE : measurable E) : - \int[(l \; k) x]_z (\1_E z)%:E = \int[l x]_y (\int[k (x, y)]_z (\1_E z)%:E). + \int[kcomp l k x]_z (\1_E z)%:E = \int[l x]_y (\int[k (x, y)]_z (\1_E z)%:E). Proof. rewrite integral_indic//= /kcomp. by apply: eq_integral => y _; rewrite integral_indic. Qed. Let integral_kcomp_nnsfun x (f : {nnsfun Z >-> R}) : - \int[(l \; k) x]_z (f z)%:E = \int[l x]_y (\int[k (x, y)]_z (f z)%:E). + \int[kcomp l k x]_z (f z)%:E = \int[l x]_y (\int[k (x, y)]_z (f z)%:E). Proof. under [in LHS]eq_integral do rewrite fimfunE -fsumEFin//. rewrite ge0_integral_fsum//; last 2 first. @@ -1099,11 +1097,11 @@ by rewrite integral0_eq// => y _; rewrite preimage_nnfun0// measure0 mule0. Qed. Lemma integral_kcomp x f : (forall z, 0 <= f z) -> measurable_fun [set: Z] f -> - \int[(l \; k) x]_z f z = \int[l x]_y (\int[k (x, y)]_z f z). + \int[kcomp l k x]_z f z = \int[l x]_y (\int[k (x, y)]_z f z). Proof. move=> f0 mf. have [f_ [ndf_ f_f]] := approximation measurableT mf (fun z _ => f0 z). -transitivity (\int[(l \; k) x]_z (lim ((f_ n z)%:E @[n --> \oo]))). +transitivity (\int[kcomp l k x]_z (lim ((f_ n z)%:E @[n --> \oo]))). by apply/eq_integral => z _; apply/esym/cvg_lim => //=; exact: f_f. rewrite monotone_convergence//; last 3 first. by move=> n; exact/EFin_measurable_fun.