Integrable On norm rpow of is Bounded compl nhds
Space.integrableOn_norm_rpow_of_isBounded_compl_nhds
Plain-language statement
The function x ↦ ‖x‖ᵖ is integrable on a bounded subset with the origin in its exterior.
Exact Lean statement
lemma integrableOn_norm_rpow_of_isBounded_compl_nhds {d : ℕ} {s : Set (Space d)}
(hs : Bornology.IsBounded s) (hs' : sᶜ ∈ nhds 0) (p : ℝ) :
IntegrableOn (fun x : Space d ↦ ‖x‖ ^ p) sFormal artifact
Lean source
lemma integrableOn_norm_rpow_of_isBounded_compl_nhds {d : ℕ} {s : Set (Space d)} (hs : Bornology.IsBounded s) (hs' : sᶜ ∈ nhds 0) (p : ℝ) : IntegrableOn (fun x : Space d ↦ ‖x‖ ^ p) s := by obtain ⟨a, ha, ha'⟩ := Metric.eventually_nhds_iff_ball.mp hs' obtain ⟨b, hb, hb'⟩ := Bornology.IsBounded.subset_ball_lt hs 0 0 have hsc : s ⊆ (Metric.ball 0 a)ᶜ := Set.subset_compl_comm.mp ha' exact IntegrableOn.mono_set (integrableOn_norm_rpow_shell ha b p) (Set.subset_inter hb' hsc)- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/SpaceAndTime/Space/Integrals/NormPow.lean:188-194
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