English
The map from the extreme points of a convex set to the ambient space is convex independent.
Русский
Отображение от крайних точек к окружающему пространству образует конвексивно независимое семейство.
LaTeX
$$$\\text{ConvexIndependent}_{\\mathbb{k}}\\bigl(\\operatorname{Subtype.val} : s_{\\text{extremePoints}} 𝕜 \\to E\\bigr)$$$
Lean4
/-- If `μ` is a non-zero finite measure on `α`, `s` is a convex closed set in `E`, and `f` is an
integrable function sending `μ`-a.e. points to `s`, then the average value of `f` belongs to `s`:
`⨍ x, f x ∂μ ∈ s`. See also `Convex.centerMass_mem` for a finite sum version of this lemma. -/
theorem set_average_mem (hs : Convex ℝ s) (hsc : IsClosed s) (h0 : μ t ≠ 0) (ht : μ t ≠ ∞)
(hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s) (hfi : IntegrableOn f t μ) : (⨍ x in t, f x ∂μ) ∈ s :=
have := Fact.mk ht.lt_top
have := NeZero.mk h0
hs.average_mem hsc hfs hfi
