Smooth Existence

@[blueprint  (title := "SmoothExistence")  (statement := /--  There exists a smooth (once differentiable would be enough),  nonnegative ``bumpfunction'' $\nu$,  supported in $[1/2,2]$ with total mass one:  $$  \int_0^\infty \nu(x)\frac{dx}{x} = 1.  $$  -/)  (proof := /-- Same idea as Urysohn-type argument. -/)]lemma SmoothExistence :    ∃ (ν : ℝ → ℝ), (ContDiff ℝ ∞ ν) ∧ (∀ x, 0 ≤ ν x) ∧    ν.support ⊆ Icc (1 / 2) 2 ∧ ∫ x in Ici 0, ν x / x = 1 := by  suffices h : ∃ (ν : ℝ → ℝ), (ContDiff ℝ ∞ ν) ∧ (∀ x, 0 ≤ ν x) ∧      ν.support ⊆ Set.Icc (1 / 2) 2 ∧ 0 < ∫ x in Set.Ici 0, ν x / x by    obtain ⟨ν, hν, hνnonneg, hνsupp, hνpos⟩ := h    let c := (∫ x in Ici 0, ν x / x)    use fun y ↦ ν y / c    refine ⟨hν.div_const c, fun y ↦ div_nonneg (hνnonneg y) (le_of_lt hνpos), ?_, ?_⟩    · rw [Function.support_div, Function.support_const (ne_of_lt hνpos).symm, inter_univ]      convert hνsupp    · simp only [div_right_comm _ c _, integral_div c, div_self <| ne_of_gt hνpos, c]  have := smooth_urysohn_support_Ioo (a := 1 / 2) (b := 1) (c := 3 / 2) (d := 2)    (by linarith) (by linarith)  obtain ⟨ν, hνContDiff, _, hν0, hν1, hνSupport⟩ := this  use ν, hνContDiff  unfold indicator at hν0 hν1  simp only [mem_Icc, Pi.one_apply, Pi.le_def, mem_Ioo] at hν0 hν1  simp only [hνSupport, subset_def, mem_Ioo, mem_Icc, and_imp]  split_ands  · exact fun x ↦ le_trans (by simp [apply_ite]) (hν0 x)  · exact fun y hy hy' ↦ ⟨by linarith, by linarith⟩  · rw [integral_pos_iff_support_of_nonneg]    · simp only [Function.support_div, measurableSet_Ici, Measure.restrict_apply',        hνSupport, Function.support_id']      have : (Ioo (1 / 2 : ℝ) 2 ∩ {0}ᶜ ∩ Ici 0) = Ioo (1 / 2) 2 := by        ext x        simp only [one_div, mem_inter_iff, mem_Ioo, mem_compl_iff, mem_singleton_iff, mem_Ici]        bound      simp only [this, volume_Ioo, ENNReal.ofReal_pos, sub_pos, gt_iff_lt]      linarith    · simp_rw [Pi.le_def, Pi.zero_apply]      intro y      by_cases h : y ∈ Function.support ν      · apply div_nonneg <| le_trans (by simp [apply_ite]) (hν0 y)        rw [hνSupport, mem_Ioo] at h        linarith [h.left]      · simp only [Function.mem_support, ne_eq, not_not] at h        simp [h]    · have : (fun x ↦ ν x / x).support ⊆ Icc (1 / 2) 2 := by        rw [Function.support_div, hνSupport]        exact (inter_subset_left).trans Ioo_subset_Icc_self      apply (integrableOn_iff_integrable_of_support_subset this).mp      apply ContinuousOn.integrableOn_compact isCompact_Icc      apply hνContDiff.continuous.continuousOn.div continuousOn_id ?_      simp only [mem_Icc, ne_eq, and_imp, id_eq]      intros; linarith