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lines changed Original file line number Diff line number Diff line change @@ -135,18 +135,19 @@ lemma measurable_densityProcess (κ : Kernel α (γ × β)) (ν : Kernel α γ)
135135lemma measurable_densityProcess_left (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ)
136136 (x : γ) {s : Set β} (hs : MeasurableSet s) :
137137 Measurable (fun a ↦ densityProcess κ ν n a x s) :=
138- (measurable_densityProcess κ ν n hs).comp (measurable_id.prodMk measurable_const)
138+ (( measurable_densityProcess κ ν n hs).comp (measurable_id.prodMk measurable_const): )
139139
140140lemma measurable_densityProcess_right (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ)
141141 {s : Set β} (a : α) (hs : MeasurableSet s) :
142142 Measurable (fun x ↦ densityProcess κ ν n a x s) :=
143- (measurable_densityProcess κ ν n hs).comp (measurable_const.prodMk measurable_id)
143+ (( measurable_densityProcess κ ν n hs).comp (measurable_const.prodMk measurable_id): )
144144
145145lemma measurable_countableFiltration_densityProcess (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ)
146146 (a : α) {s : Set β} (hs : MeasurableSet s) :
147147 Measurable[countableFiltration γ n] (fun x ↦ densityProcess κ ν n a x s) := by
148148 refine @Measurable.ennreal_toReal _ (countableFiltration γ n) _ ?_
149- exact (measurable_densityProcess_countableFiltration_aux κ ν n hs).comp measurable_prodMk_left
149+ -- The exact also works without the `( :)`, but is a bit slow.
150+ exact ((measurable_densityProcess_countableFiltration_aux κ ν n hs).comp measurable_prodMk_left :)
150151
151152lemma stronglyMeasurable_countableFiltration_densityProcess (κ : Kernel α (γ × β)) (ν : Kernel α γ)
152153 (n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) :
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