English
A convex independent subset still holds when you remove a point and look at the remainder.
Русский
Если подмножество выпукво независимое, то при удалении точки независимость сохраняется для оставшейся части.
LaTeX
$$$\\text{ConvexIndependent}_{\\mathbb{k}}((\\uparrow):s\\to E) \\Rightarrow \\text{ConvexIndependent}_{\\mathbb{k}}((\\uparrow):s\\setminus\\{x\\}\\to E)$$$
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 average_mem [IsFiniteMeasure μ] [NeZero μ] (hs : Convex ℝ s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
(hfi : Integrable f μ) : (⨍ x, f x ∂μ) ∈ s :=
hs.integral_mem hsc (ae_mono' smul_absolutelyContinuous hfs) hfi.to_average
