integral_antitoneOn_of_integrand_ae

English

If the integrand f(x,a) is convex in a and the integrals of f(x,·) are integrable, then the map a ↦ ∫ f(x,a) dμ is convex on the domain.

Русский

Если f(x,a) выпукло по a и интегралы интегрируемы, то a ↦ ∫ f(x,a) dμ выпуклa на области.

LaTeX

$$$ConvexOn(a) \\Rightarrow ConvexOn(a) (\\int f(x,a) dμ)$.$$

Lean4

theorem integral_antitoneOn_of_integrand_ae {β : Type*} [Preorder β] {f : α → β → E} {s : Set β}
    (hf_anti : ∀ᵐ x ∂μ, AntitoneOn (f x) s) (hf_int : ∀ a ∈ s, Integrable (f · a) μ) :
    AntitoneOn (fun b => ∫ x, f x b ∂μ) s := by
  intro a ha b hb hab
  refine integral_mono_ae (hf_int b hb) (hf_int a ha) ?_
  filter_upwards [hf_anti] with x hx
  exact hx ha hb hab

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