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12 changes: 1 addition & 11 deletions Mathlib/Probability/Distributions/Gaussian/Fernique.lean
Original file line number Diff line number Diff line change
Expand Up @@ -152,17 +152,7 @@ lemma integrable_exp_sq_of_conv_neg (μ : Measure E) [IsGaussian μ] {C C' : ℝ
_ = (1 + ε) * ‖x - y‖ ^ 2 + (1 + 1 / ε) * ‖y‖ ^ 2 := by ring
calc C' * ‖x‖ ^ 2
_ ≤ C' * ((1 + ε) * ‖x - y‖ ^ 2 + (1 + 1 / ε) * ‖y‖ ^ 2) := by gcongr
_ = (C' * (1 + 1 / ε)) * ‖y‖ ^ 2 + (C' * (1 + ε)) * ‖x - y‖ ^ 2 := by ring
_ = C / ε * ‖y‖ ^ 2 + C * ‖x - y‖ ^ 2 := by
unfold ε
congr 2
· simp only [one_div, inv_div]
rw [one_add_div (by rw [sub_ne_zero]; exact hC'_lt.ne'), div_div_eq_mul_div]
simp only [sub_add_cancel]
ring
· rw [one_add_div (by positivity)]
simp only [add_sub_cancel]
rw [mul_div_cancel₀ _ (by positivity)]
_ = C / ε * ‖y‖ ^ 2 + C * ‖x - y‖ ^ 2 := by grind

/-- **Fernique's theorem**: for a Gaussian measure, there exists `C > 0` such that the function
`x ↦ exp (C * ‖x‖ ^ 2)` is integrable. -/
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