English
A fiberwise Theta property holds: if f =Θ[l×ˢ l'] g, then for almost every y in l' the function x ↦ f(x,y) is Θ_l with g(x,y).
Русский
Свойство Θ по срезам слева: если f =Θ[l×ˢ l'] g, то для почти всех y в l' срезы удовлетворяют Θ по x.
LaTeX
$$$\forall f,g:\; f =Θ[\mathrm{Filter.instSProd.sprod}(l,l')] g \Rightarrow \forall^\infty y\in l',\; f(\cdot,y) =Θ_l (g(\cdot,y))$$$
Lean4
/-- A real function into a finite-dimensional real vector space with bounded variation
is differentiable almost everywhere. -/
theorem ae_differentiableAt {f : ℝ → V} (h : LocallyBoundedVariationOn f univ) : ∀ᵐ x, DifferentiableAt ℝ f x :=
by
filter_upwards [h.ae_differentiableWithinAt_of_mem] with x hx
rw [differentiableWithinAt_univ] at hx
exact hx (mem_univ _)