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ef96466 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.RadialPressureMeasure
namespace OAI
/-! The angular-gradient radial measure in dimension twelve. -/
open Set MeasureTheory
namespace DefocusingNLS
noncomputable def spectralAngularMeasure (R : ℝ) : Measure ℝ :=
(volume.restrict (Icc 0 R)).withDensity (fun r : ℝ => ENNReal.ofReal ((max r 0)^9))
instance spectralAngularMeasure_finite (R : ℝ) : IsFiniteMeasure (spectralAngularMeasure R) := by
have hc : Continuous (fun r : ℝ => (max r 0)^9) := by fun_prop
have hi := hc.continuousOn.integrableOn_compact (μ := volume) (K := Icc 0 R) isCompact_Icc
apply isFiniteMeasure_withDensity
exact (lintegral_ofReal_ne_top_iff_integrable hi.aestronglyMeasurable
(Filter.Eventually.of_forall (fun r => by positivity))).2 hi
theorem spectralAngularMeasure_integral (R : ℝ) (hR : 0 ≤ R) (f : ℝ → ℝ) :
(∫ r, f r ∂spectralAngularMeasure R)=∫ r in (0 : ℝ)..R, f r*r^9 := by
unfold spectralAngularMeasure
rw [integral_withDensity_eq_integral_toReal_smul (by fun_prop)
(Filter.Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top)),intervalIntegral.integral_of_le hR]
calc
_ = ∫ r in Icc (0 : ℝ) R, f r*r^9 := by
apply setIntegral_congr_fun measurableSet_Icc
intro r hr
change (ENNReal.ofReal ((max r 0)^9)).toReal • f r=f r*r^9
rw [ENNReal.toReal_ofReal (by positivity),max_eq_left hr.1]
simp only [smul_eq_mul]
ring
_ = _ := integral_Icc_eq_integral_Ioc
end DefocusingNLS
end OAI
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