| import Mathlib |
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| namespace OAI |
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| namespace Erdos970 |
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| section |
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| open MeasureTheory Set |
| namespace ErdosContinuousBoundary |
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| 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 |
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| end ErdosContinuousBoundary |
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| end |
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| end Erdos970 |
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| end OAI |
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