English
If derivative is bounded by NNReal C, then LipschitzWith C holds.
Русский
Если производная ограничена NNReal C, то LipschitzWith C держится.
LaTeX
$$$$ \text{lipschitz with NNReal constant } C $$$$
Lean4
/-- A Stieltjes function is almost everywhere differentiable, with derivative equal to the
Radon-Nikodym derivative of the associated Stieltjes measure with respect to Lebesgue. -/
theorem ae_hasDerivAt (f : StieltjesFunction) : ∀ᵐ x, HasDerivAt f (rnDeriv f.measure volume x).toReal x := by
/- Denote by `μ` the Stieltjes measure associated to `f`.
The general theorem `VitaliFamily.ae_tendsto_rnDeriv` ensures that `μ [x, y] / (y - x)` tends
to the Radon-Nikodym derivative as `y` tends to `x` from the right. As `μ [x,y] = f y - f (x^-)`
and `f (x^-) = f x` almost everywhere, this gives differentiability on the right.
On the left, `μ [y, x] / (x - y)` again tends to the Radon-Nikodym derivative.
As `μ [y, x] = f x - f (y^-)`, this is not exactly the right result, so one uses a sandwiching
argument to deduce the convergence for `(f x - f y) / (x - y)`. -/
filter_upwards [VitaliFamily.ae_tendsto_rnDeriv (vitaliFamily (volume : Measure ℝ) 1) f.measure,
rnDeriv_lt_top f.measure volume, f.countable_leftLim_ne.ae_notMem volume] with x hx h'x h''x
have L1 : Tendsto (fun y => (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (rnDeriv f.measure volume x).toReal) :=
by
apply Tendsto.congr' _ ((ENNReal.tendsto_toReal h'x.ne).comp (hx.comp (Real.tendsto_Icc_vitaliFamily_right x)))
filter_upwards [self_mem_nhdsWithin]
rintro y (hxy : x < y)
simp only [comp_apply, StieltjesFunction.measure_Icc, Real.volume_Icc, Classical.not_not.1 h''x]
rw [← ENNReal.ofReal_div_of_pos (sub_pos.2 hxy), ENNReal.toReal_ofReal]
exact div_nonneg (sub_nonneg.2 (f.mono hxy.le)) (sub_pos.2 hxy).le
have L2 : Tendsto (fun y => (leftLim f y - f x) / (y - x)) (𝓝[<] x) (𝓝 (rnDeriv f.measure volume x).toReal) :=
by
apply Tendsto.congr' _ ((ENNReal.tendsto_toReal h'x.ne).comp (hx.comp (Real.tendsto_Icc_vitaliFamily_left x)))
filter_upwards [self_mem_nhdsWithin]
rintro y (hxy : y < x)
simp only [comp_apply, StieltjesFunction.measure_Icc, Real.volume_Icc]
rw [← ENNReal.ofReal_div_of_pos (sub_pos.2 hxy), ENNReal.toReal_ofReal, ← neg_neg (y - x), div_neg, neg_div',
neg_sub, neg_sub]
exact div_nonneg (sub_nonneg.2 (f.mono.leftLim_le hxy.le)) (sub_pos.2 hxy).le
have L3 :
Tendsto (fun y => (leftLim f (y + 1 * (y - x) ^ 2) - f x) / (y - x)) (𝓝[<] x)
(𝓝 (rnDeriv f.measure volume x).toReal) :=
by
apply tendsto_apply_add_mul_sq_div_sub (nhdsLT_le_nhdsNE x) L2
apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within
· apply Tendsto.mono_left _ nhdsWithin_le_nhds
have : Tendsto (fun y : ℝ => y + ↑1 * (y - x) ^ 2) (𝓝 x) (𝓝 (x + ↑1 * (x - x) ^ 2)) :=
tendsto_id.add (((tendsto_id.sub_const x).pow 2).const_mul ↑1)
simpa using this
· filter_upwards [Ioo_mem_nhdsLT <| show x - 1 < x by simp]
rintro y ⟨hy : x - 1 < y, h'y : y < x⟩
rw [mem_Iio]
nlinarith
-- Deduce the correct limit on the left, by sandwiching.
have L4 : Tendsto (fun y => (f y - f x) / (y - x)) (𝓝[<] x) (𝓝 (rnDeriv f.measure volume x).toReal) :=
by
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' L3 L2
· filter_upwards [self_mem_nhdsWithin]
rintro y (hy : y < x)
refine div_le_div_of_nonpos_of_le (by linarith) ((sub_le_sub_iff_right _).2 ?_)
apply f.mono.le_leftLim
have : ↑0 < (x - y) ^ 2 := sq_pos_of_pos (sub_pos.2 hy)
linarith
· filter_upwards [self_mem_nhdsWithin]
rintro y (hy : y < x)
refine div_le_div_of_nonpos_of_le (by linarith) ?_
simpa only [sub_le_sub_iff_right] using
f.mono.leftLim_le
(le_refl y)
-- prove the result by splitting into left and right limits.
rw [hasDerivAt_iff_tendsto_slope, slope_fun_def_field, ← nhdsLT_sup_nhdsGT, tendsto_sup]
exact ⟨L4, L1⟩
