integrable_exp_mul_of_nonpos_of_ge

English

Under MemLp id 2 μ, the covariance bilinear dual equals the integral of L1(x - μ[id]) L2(x - μ[id]) over μ.

Русский

При MemLp id 2 μ ковариационная билинейная двойка равна интегралу L1(x - μ[id]) L2(x - μ[id]) по μ.

LaTeX

$$$covarianceBilinDual(\\mu) L_1 L_2 = \\int_E L_1(x - μ[id]) L_2(x - μ[id]) \\, d\\mu(x)$$$

Lean4

/-- If `ω ↦ exp (u * X ω)` is integrable at `u ≤ 0`, then it is integrable on `[u, 0]`. -/
theorem integrable_exp_mul_of_nonpos_of_ge [IsFiniteMeasure μ] (hu : Integrable (fun ω ↦ exp (u * X ω)) μ)
    (h_nonpos : t ≤ 0) (htu : u ≤ t) : Integrable (fun ω ↦ exp (t * X ω)) μ :=
  integrable_exp_mul_of_le_of_le hu (by simp) htu h_nonpos

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