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f89a930 | 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 | import Mathlib
namespace OAI
namespace Erdos970
section
open MeasureTheory Set
namespace ErdosContinuousBoundary
theorem continuous_variable_upper_integral
(F : (ℝ×ℝ) → ℝ → ℝ) (hF : Continuous F.uncurry)
(c : (ℝ×ℝ) → ℝ) (hc : Continuous c) :
Continuous (fun p => ∫ x in (1:ℝ)..c p,F p x) := by
let G : (ℝ×ℝ) → ℝ → ℝ := fun p t => F p ((c p-1)*t+1)
have hG : Continuous G.uncurry := by
change Continuous (F.uncurry ∘ (fun q : (ℝ×ℝ)×ℝ => (q.1,(c q.1-1)*q.2+1)))
apply hF.comp
exact continuous_fst.prodMk
(((hc.comp continuous_fst).sub continuous_const).mul continuous_snd |>.add continuous_const)
have hi : Continuous (fun p => ∫ t in Icc (0:ℝ) 1,G p t) :=
continuous_parametric_integral_of_continuous hG isCompact_Icc
have heq (p : ℝ×ℝ) : (c p-1)*(∫ t in Icc (0:ℝ) 1,G p t)=∫ x in (1:ℝ)..c p,F p x := by
have h := intervalIntegral.smul_integral_comp_mul_add (a:=0) (b:=1) (fun x => F p x) (c p-1) 1
rw [intervalIntegral.integral_of_le (by norm_num : (0:ℝ)≤1),← integral_Icc_eq_integral_Ioc] at h
simpa only [smul_eq_mul,mul_zero,zero_add,mul_one,sub_add_cancel,G] using h
exact ((hc.sub continuous_const).mul hi).congr heq
end ErdosContinuousBoundary
end
end Erdos970
end OAI
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