English
Uniform inducing maps preserve uniform equicontinuity on a set: F is UniformEquicontinuousOn S iff the postcomposed family is.
Русский
Уравномерно индуцирующие отображения сохраняют uniforme экквинтоентность на множество: UniformEquicontinuousOn S ⇔ посткомпозицией полученная семья экквинтично.
LaTeX
$$$\\operatorname{UniformEquicontinuousOn} F S \\iff \\operatorname{UniformEquicontinuousOn} ((u \\circ \\cdot) \\circ F) S.$$$
Lean4
/-- Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous
on a subset `S : Set β` iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`,
is uniformly equicontinuous on `S`. -/
theorem uniformEquicontinuousOn_iff {F : ι → β → α} {S : Set β} {u : α → γ} (hu : IsUniformInducing u) :
UniformEquicontinuousOn F S ↔ UniformEquicontinuousOn ((u ∘ ·) ∘ F) S :=
by
have := UniformFun.postcomp_isUniformInducing (α := ι) hu
simp only [uniformEquicontinuousOn_iff_uniformContinuousOn, this.uniformContinuousOn_iff]
rfl
