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[Merged by Bors] - feat(MeasureTheory): add MeasurableEmbedding.natCast #34521

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huaizhangchu wants to merge 5 commits into from
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[Merged by Bors] - feat(MeasureTheory): add MeasurableEmbedding.natCast #34521

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435873c
feat(MeasureTheory): add measurableEmbedding_natCast
vianna04 Jan 28, 2026
e46b272
Merge branch 'master' into measurableEmbedding-natCast
huaizhangchu Jan 28, 2026
26f7455
Merge branch 'master' into measurableEmbedding-natCast
huaizhangchu Mar 3, 2026
a7fc251
generalize measurableEmbedding_natCast beyond Real
vianna04 Mar 3, 2026
5740487
move measurableEmbedding_natCast to Embedding.lean
vianna04 Mar 4, 2026
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9 changes: 9 additions & 0 deletions Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean
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Expand Up @@ -153,6 +153,15 @@ theorem AEMeasurable.real_toNNReal {f : α → ℝ} {μ : Measure α} (hf : AEMe
AEMeasurable (fun x => Real.toNNReal (f x)) μ :=
measurable_real_toNNReal.comp_aemeasurable hf

lemma measurableEmbedding_natCast {α : Type*} [MeasurableSpace α]
[MeasurableSingletonClass α] [AddMonoidWithOne α] [CharZero α] :
MeasurableEmbedding (Nat.cast : ℕ → α) where
injective := CharZero.cast_injective
measurable := measurable_from_nat
measurableSet_image' := fun _ _ =>
((Set.countable_range (Nat.cast : ℕ → α)).mono
(Set.image_subset_range _ _)).measurableSet

theorem measurable_coe_nnreal_real : Measurable ((↑) : ℝ≥0 → ℝ) :=
NNReal.continuous_coe.measurable

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