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jacobian.lean
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/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import measure_theory.covering.besicovitch_vector_space
import measure_theory.measure.haar_lebesgue
import analysis.normed_space.pointwise
import measure_theory.constructions.polish
/-!
# Change of variables in higher-dimensional integrals
Let `μ` be a Lebesgue measure on a finite-dimensional real vector space `E`.
Let `f : E → E` be a function which is injective and differentiable on a measurable set `s`,
with derivative `f'`. Then we prove that `f '' s` is measurable, and
its measure is given by the formula `μ (f '' s) = ∫⁻ x in s, |(f' x).det| ∂μ` (where `(f' x).det`
is almost everywhere measurable, but not Borel-measurable in general). This formula is proved in
`lintegral_abs_det_fderiv_eq_add_haar_image`. We deduce the change of variables
formula for the Lebesgue and Bochner integrals, in `lintegral_image_eq_lintegral_abs_det_fderiv_mul`
and `integral_image_eq_integral_abs_det_fderiv_smul` respectively.
## Main results
* `add_haar_image_eq_zero_of_differentiable_on_of_add_haar_eq_zero`: if `f` is differentiable on a
set `s` with zero measure, then `f '' s` also has zero measure.
* `add_haar_image_eq_zero_of_det_fderiv_within_eq_zero`: if `f` is differentiable on a set `s`, and
its derivative is never invertible, then `f '' s` has zero measure (a version of Sard's lemma).
* `ae_measurable_fderiv_within`: if `f` is differentiable on a measurable set `s`, then `f'`
is almost everywhere measurable on `s`.
For the next statements, `s` is a measurable set and `f` is differentiable on `s`
(with a derivative `f'`) and injective on `s`.
* `measurable_image_of_fderiv_within`: the image `f '' s` is measurable.
* `measurable_embedding_of_fderiv_within`: the function `s.restrict f` is a measurable embedding.
* `lintegral_abs_det_fderiv_eq_add_haar_image`: the image measure is given by
`μ (f '' s) = ∫⁻ x in s, |(f' x).det| ∂μ`.
* `lintegral_image_eq_lintegral_abs_det_fderiv_mul`: for `g : E → ℝ≥0∞`, one has
`∫⁻ x in f '' s, g x ∂μ = ∫⁻ x in s, ennreal.of_real (|(f' x).det|) * g (f x) ∂μ`.
* `integral_image_eq_integral_abs_det_fderiv_smul`: for `g : E → F`, one has
`∫ x in f '' s, g x ∂μ = ∫ x in s, |(f' x).det| • g (f x) ∂μ`.
* `integrable_on_image_iff_integrable_on_abs_det_fderiv_smul`: for `g : E → F`, the function `g` is
integrable on `f '' s` if and only if `|(f' x).det| • g (f x))` is integrable on `s`.
## Implementation
Typical versions of these results in the literature have much stronger assumptions: `s` would
typically be open, and the derivative `f' x` would depend continuously on `x` and be invertible
everywhere, to have the local inverse theorem at our disposal. The proof strategy under our weaker
assumptions is more involved. We follow [Fremlin, *Measure Theory* (volume 2)][fremlin_vol2].
The first remark is that, if `f` is sufficiently well approximated by a linear map `A` on a set
`s`, then `f` expands the volume of `s` by at least `A.det - ε` and at most `A.det + ε`, where
the closeness condition depends on `A` in a non-explicit way (see `add_haar_image_le_mul_of_det_lt`
and `mul_le_add_haar_image_of_lt_det`). This fact holds for balls by a simple inclusion argument,
and follows for general sets using the Besicovitch covering theorem to cover the set by balls with
measures adding up essentially to `μ s`.
When `f` is differentiable on `s`, one may partition `s` into countably many subsets `s ∩ t n`
(where `t n` is measurable), on each of which `f` is well approximated by a linear map, so that the
above results apply. See `exists_partition_approximates_linear_on_of_has_fderiv_within_at`, which
follows from the pointwise differentiability (in a non-completely trivial way, as one should ensure
a form of uniformity on the sets of the partition).
Combining the above two results would give the conclusion, except for two difficulties: it is not
obvious why `f '' s` and `f'` should be measurable, which prevents us from using countable
additivity for the measure and the integral. It turns out that `f '' s` is indeed measurable,
and that `f'` is almost everywhere measurable, which is enough to recover countable additivity.
The measurability of `f '' s` follows from the deep Lusin-Souslin theorem ensuring that, in a
Polish space, a continuous injective image of a measurable set is measurable.
The key point to check the almost everywhere measurability of `f'` is that, if `f` is approximated
up to `δ` by a linear map on a set `s`, then `f'` is within `δ` of `A` on a full measure subset
of `s` (namely, its density points). With the above approximation argument, it follows that `f'`
is the almost everywhere limit of a sequence of measurable functions (which are constant on the
pieces of the good discretization), and is therefore almost everywhere measurable.
## Tags
Change of variables in integrals
## References
[Fremlin, *Measure Theory* (volume 2)][fremlin_vol2]
-/
open measure_theory measure_theory.measure metric filter set finite_dimensional asymptotics
topological_space
open_locale nnreal ennreal topology pointwise
variables {E F : Type*} [normed_add_comm_group E] [normed_space ℝ E] [finite_dimensional ℝ E]
[normed_add_comm_group F] [normed_space ℝ F] {s : set E} {f : E → E} {f' : E → E →L[ℝ] E}
/-!
### Decomposition lemmas
We state lemmas ensuring that a differentiable function can be approximated, on countably many
measurable pieces, by linear maps (with a prescribed precision depending on the linear map).
-/
/-- Assume that a function `f` has a derivative at every point of a set `s`. Then one may cover `s`
with countably many closed sets `t n` on which `f` is well approximated by linear maps `A n`. -/
lemma exists_closed_cover_approximates_linear_on_of_has_fderiv_within_at
[second_countable_topology F]
(f : E → F) (s : set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x)
(r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
∃ (t : ℕ → set E) (A : ℕ → (E →L[ℝ] F)), (∀ n, is_closed (t n)) ∧ (s ⊆ ⋃ n, t n)
∧ (∀ n, approximates_linear_on f (A n) (s ∩ t n) (r (A n)))
∧ (s.nonempty → ∀ n, ∃ y ∈ s, A n = f' y) :=
begin
/- Choose countably many linear maps `f' z`. For every such map, if `f` has a derivative at `x`
close enough to `f' z`, then `f y - f x` is well approximated by `f' z (y - x)` for `y` close
enough to `x`, say on a ball of radius `r` (or even `u n` for some `n`, where `u` is a fixed
sequence tending to `0`).
Let `M n z` be the points where this happens. Then this set is relatively closed inside `s`,
and moreover in every closed ball of radius `u n / 3` inside it the map is well approximated by
`f' z`. Using countably many closed balls to split `M n z` into small diameter subsets `K n z p`,
one obtains the desired sets `t q` after reindexing.
-/
-- exclude the trivial case where `s` is empty
rcases eq_empty_or_nonempty s with rfl|hs,
{ refine ⟨λ n, ∅, λ n, 0, _, _, _, _⟩;
simp },
-- we will use countably many linear maps. Select these from all the derivatives since the
-- space of linear maps is second-countable
obtain ⟨T, T_count, hT⟩ : ∃ T : set s, T.countable ∧
(⋃ x ∈ T, ball (f' (x : E)) (r (f' x))) = ⋃ (x : s), ball (f' x) (r (f' x)) :=
topological_space.is_open_Union_countable _ (λ x, is_open_ball),
-- fix a sequence `u` of positive reals tending to zero.
obtain ⟨u, u_anti, u_pos, u_lim⟩ :
∃ (u : ℕ → ℝ), strict_anti u ∧ (∀ (n : ℕ), 0 < u n) ∧ tendsto u at_top (𝓝 0) :=
exists_seq_strict_anti_tendsto (0 : ℝ),
-- `M n z` is the set of points `x` such that `f y - f x` is close to `f' z (y - x)` for `y`
-- in the ball of radius `u n` around `x`.
let M : ℕ → T → set E := λ n z, {x | x ∈ s ∧
∀ y ∈ s ∩ ball x (u n), ‖f y - f x - f' z (y - x)‖ ≤ r (f' z) * ‖y - x‖},
-- As `f` is differentiable everywhere on `s`, the sets `M n z` cover `s` by design.
have s_subset : ∀ x ∈ s, ∃ (n : ℕ) (z : T), x ∈ M n z,
{ assume x xs,
obtain ⟨z, zT, hz⟩ : ∃ z ∈ T, f' x ∈ ball (f' (z : E)) (r (f' z)),
{ have : f' x ∈ ⋃ (z ∈ T), ball (f' (z : E)) (r (f' z)),
{ rw hT,
refine mem_Union.2 ⟨⟨x, xs⟩, _⟩,
simpa only [mem_ball, subtype.coe_mk, dist_self] using (rpos (f' x)).bot_lt },
rwa mem_Union₂ at this },
obtain ⟨ε, εpos, hε⟩ : ∃ (ε : ℝ), 0 < ε ∧ ‖f' x - f' z‖ + ε ≤ r (f' z),
{ refine ⟨r (f' z) - ‖f' x - f' z‖, _, le_of_eq (by abel)⟩,
simpa only [sub_pos] using mem_ball_iff_norm.mp hz },
obtain ⟨δ, δpos, hδ⟩ : ∃ (δ : ℝ) (H : 0 < δ),
ball x δ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖} :=
metric.mem_nhds_within_iff.1 (is_o.def (hf' x xs) εpos),
obtain ⟨n, hn⟩ : ∃ n, u n < δ := ((tendsto_order.1 u_lim).2 _ δpos).exists,
refine ⟨n, ⟨z, zT⟩, ⟨xs, _⟩⟩,
assume y hy,
calc ‖f y - f x - (f' z) (y - x)‖
= ‖(f y - f x - (f' x) (y - x)) + (f' x - f' z) (y - x)‖ :
begin
congr' 1,
simp only [continuous_linear_map.coe_sub', map_sub, pi.sub_apply],
abel,
end
... ≤ ‖f y - f x - (f' x) (y - x)‖ + ‖(f' x - f' z) (y - x)‖ : norm_add_le _ _
... ≤ ε * ‖y - x‖ + ‖f' x - f' z‖ * ‖y - x‖ :
begin
refine add_le_add (hδ _) (continuous_linear_map.le_op_norm _ _),
rw inter_comm,
exact inter_subset_inter_right _ (ball_subset_ball hn.le) hy,
end
... ≤ r (f' z) * ‖y - x‖ :
begin
rw [← add_mul, add_comm],
exact mul_le_mul_of_nonneg_right hε (norm_nonneg _),
end },
-- the sets `M n z` are relatively closed in `s`, as all the conditions defining it are clearly
-- closed
have closure_M_subset : ∀ n z, s ∩ closure (M n z) ⊆ M n z,
{ rintros n z x ⟨xs, hx⟩,
refine ⟨xs, λ y hy, _⟩,
obtain ⟨a, aM, a_lim⟩ : ∃ (a : ℕ → E), (∀ k, a k ∈ M n z) ∧ tendsto a at_top (𝓝 x) :=
mem_closure_iff_seq_limit.1 hx,
have L1 : tendsto (λ (k : ℕ), ‖f y - f (a k) - (f' z) (y - a k)‖) at_top
(𝓝 ‖f y - f x - (f' z) (y - x)‖),
{ apply tendsto.norm,
have L : tendsto (λ k, f (a k)) at_top (𝓝 (f x)),
{ apply (hf' x xs).continuous_within_at.tendsto.comp,
apply tendsto_nhds_within_of_tendsto_nhds_of_eventually_within _ a_lim,
exact eventually_of_forall (λ k, (aM k).1) },
apply tendsto.sub (tendsto_const_nhds.sub L),
exact ((f' z).continuous.tendsto _).comp (tendsto_const_nhds.sub a_lim) },
have L2 : tendsto (λ (k : ℕ), (r (f' z) : ℝ) * ‖y - a k‖) at_top (𝓝 (r (f' z) * ‖y - x‖)) :=
(tendsto_const_nhds.sub a_lim).norm.const_mul _,
have I : ∀ᶠ k in at_top, ‖f y - f (a k) - (f' z) (y - a k)‖ ≤ r (f' z) * ‖y - a k‖,
{ have L : tendsto (λ k, dist y (a k)) at_top (𝓝 (dist y x)) := tendsto_const_nhds.dist a_lim,
filter_upwards [(tendsto_order.1 L).2 _ hy.2],
assume k hk,
exact (aM k).2 y ⟨hy.1, hk⟩ },
exact le_of_tendsto_of_tendsto L1 L2 I },
-- choose a dense sequence `d p`
rcases topological_space.exists_dense_seq E with ⟨d, hd⟩,
-- split `M n z` into subsets `K n z p` of small diameters by intersecting with the ball
-- `closed_ball (d p) (u n / 3)`.
let K : ℕ → T → ℕ → set E := λ n z p, closure (M n z) ∩ closed_ball (d p) (u n / 3),
-- on the sets `K n z p`, the map `f` is well approximated by `f' z` by design.
have K_approx : ∀ n (z : T) p, approximates_linear_on f (f' z) (s ∩ K n z p) (r (f' z)),
{ assume n z p x hx y hy,
have yM : y ∈ M n z := closure_M_subset _ _ ⟨hy.1, hy.2.1⟩,
refine yM.2 _ ⟨hx.1, _⟩,
calc dist x y ≤ dist x (d p) + dist y (d p) : dist_triangle_right _ _ _
... ≤ u n / 3 + u n / 3 : add_le_add hx.2.2 hy.2.2
... < u n : by linarith [u_pos n] },
-- the sets `K n z p` are also closed, again by design.
have K_closed : ∀ n (z : T) p, is_closed (K n z p) :=
λ n z p, is_closed_closure.inter is_closed_ball,
-- reindex the sets `K n z p`, to let them only depend on an integer parameter `q`.
obtain ⟨F, hF⟩ : ∃ F : ℕ → ℕ × T × ℕ, function.surjective F,
{ haveI : encodable T := T_count.to_encodable,
haveI : nonempty T,
{ unfreezingI { rcases eq_empty_or_nonempty T with rfl|hT },
{ rcases hs with ⟨x, xs⟩,
rcases s_subset x xs with ⟨n, z, hnz⟩,
exact false.elim z.2 },
{ exact hT.coe_sort } },
inhabit (ℕ × T × ℕ),
exact ⟨_, encodable.surjective_decode_iget _⟩ },
-- these sets `t q = K n z p` will do
refine ⟨λ q, K (F q).1 (F q).2.1 (F q).2.2, λ q, f' (F q).2.1, λ n, K_closed _ _ _, λ x xs, _,
λ q, K_approx _ _ _, λ h's q, ⟨(F q).2.1, (F q).2.1.1.2, rfl⟩⟩,
-- the only fact that needs further checking is that they cover `s`.
-- we already know that any point `x ∈ s` belongs to a set `M n z`.
obtain ⟨n, z, hnz⟩ : ∃ (n : ℕ) (z : T), x ∈ M n z := s_subset x xs,
-- by density, it also belongs to a ball `closed_ball (d p) (u n / 3)`.
obtain ⟨p, hp⟩ : ∃ (p : ℕ), x ∈ closed_ball (d p) (u n / 3),
{ have : set.nonempty (ball x (u n / 3)),
{ simp only [nonempty_ball], linarith [u_pos n] },
obtain ⟨p, hp⟩ : ∃ (p : ℕ), d p ∈ ball x (u n / 3) := hd.exists_mem_open is_open_ball this,
exact ⟨p, (mem_ball'.1 hp).le⟩ },
-- choose `q` for which `t q = K n z p`.
obtain ⟨q, hq⟩ : ∃ q, F q = (n, z, p) := hF _,
-- then `x` belongs to `t q`.
apply mem_Union.2 ⟨q, _⟩,
simp only [hq, subset_closure hnz, hp, mem_inter_iff, and_self],
end
variables [measurable_space E] [borel_space E] (μ : measure E) [is_add_haar_measure μ]
/-- Assume that a function `f` has a derivative at every point of a set `s`. Then one may
partition `s` into countably many disjoint relatively measurable sets (i.e., intersections
of `s` with measurable sets `t n`) on which `f` is well approximated by linear maps `A n`. -/
lemma exists_partition_approximates_linear_on_of_has_fderiv_within_at
[second_countable_topology F]
(f : E → F) (s : set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x)
(r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
∃ (t : ℕ → set E) (A : ℕ → (E →L[ℝ] F)), pairwise (disjoint on t)
∧ (∀ n, measurable_set (t n)) ∧ (s ⊆ ⋃ n, t n)
∧ (∀ n, approximates_linear_on f (A n) (s ∩ t n) (r (A n)))
∧ (s.nonempty → ∀ n, ∃ y ∈ s, A n = f' y) :=
begin
rcases exists_closed_cover_approximates_linear_on_of_has_fderiv_within_at f s f' hf' r rpos
with ⟨t, A, t_closed, st, t_approx, ht⟩,
refine ⟨disjointed t, A, disjoint_disjointed _,
measurable_set.disjointed (λ n, (t_closed n).measurable_set), _, _, ht⟩,
{ rw Union_disjointed, exact st },
{ assume n, exact (t_approx n).mono_set (inter_subset_inter_right _ (disjointed_subset _ _)) },
end
namespace measure_theory
/-!
### Local lemmas
We check that a function which is well enough approximated by a linear map expands the volume
essentially like this linear map, and that its derivative (if it exists) is almost everywhere close
to the approximating linear map.
-/
/-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at most `m` for any `m > det A`. -/
lemma add_haar_image_le_mul_of_det_lt
(A : E →L[ℝ] E) {m : ℝ≥0} (hm : ennreal.of_real (|A.det|) < m) :
∀ᶠ δ in 𝓝[>] (0 : ℝ≥0), ∀ (s : set E) (f : E → E) (hf : approximates_linear_on f A s δ),
μ (f '' s) ≤ m * μ s :=
begin
apply nhds_within_le_nhds,
let d := ennreal.of_real (|A.det|),
-- construct a small neighborhood of `A '' (closed_ball 0 1)` with measure comparable to
-- the determinant of `A`.
obtain ⟨ε, hε, εpos⟩ : ∃ (ε : ℝ),
μ (closed_ball 0 ε + A '' (closed_ball 0 1)) < m * μ (closed_ball 0 1) ∧ 0 < ε,
{ have HC : is_compact (A '' closed_ball 0 1) :=
(proper_space.is_compact_closed_ball _ _).image A.continuous,
have L0 : tendsto (λ ε, μ (cthickening ε (A '' (closed_ball 0 1))))
(𝓝[>] 0) (𝓝 (μ (A '' (closed_ball 0 1)))),
{ apply tendsto.mono_left _ nhds_within_le_nhds,
exact tendsto_measure_cthickening_of_is_compact HC },
have L1 : tendsto (λ ε, μ (closed_ball 0 ε + A '' (closed_ball 0 1)))
(𝓝[>] 0) (𝓝 (μ (A '' (closed_ball 0 1)))),
{ apply L0.congr' _,
filter_upwards [self_mem_nhds_within] with r hr,
rw [←HC.add_closed_ball_zero (le_of_lt hr), add_comm] },
have L2 : tendsto (λ ε, μ (closed_ball 0 ε + A '' (closed_ball 0 1)))
(𝓝[>] 0) (𝓝 (d * μ (closed_ball 0 1))),
{ convert L1,
exact (add_haar_image_continuous_linear_map _ _ _).symm },
have I : d * μ (closed_ball 0 1) < m * μ (closed_ball 0 1) :=
(ennreal.mul_lt_mul_right ((measure_closed_ball_pos μ _ zero_lt_one).ne')
measure_closed_ball_lt_top.ne).2 hm,
have H : ∀ᶠ (b : ℝ) in 𝓝[>] 0,
μ (closed_ball 0 b + A '' closed_ball 0 1) < m * μ (closed_ball 0 1) :=
(tendsto_order.1 L2).2 _ I,
exact (H.and self_mem_nhds_within).exists },
have : Iio (⟨ε, εpos.le⟩ : ℝ≥0) ∈ 𝓝 (0 : ℝ≥0), { apply Iio_mem_nhds, exact εpos },
filter_upwards [this],
-- fix a function `f` which is close enough to `A`.
assume δ hδ s f hf,
-- This function expands the volume of any ball by at most `m`
have I : ∀ x r, x ∈ s → 0 ≤ r → μ (f '' (s ∩ closed_ball x r)) ≤ m * μ (closed_ball x r),
{ assume x r xs r0,
have K : f '' (s ∩ closed_ball x r) ⊆ A '' (closed_ball 0 r) + closed_ball (f x) (ε * r),
{ rintros y ⟨z, ⟨zs, zr⟩, rfl⟩,
apply set.mem_add.2 ⟨A (z - x), f z - f x - A (z - x) + f x, _, _, _⟩,
{ apply mem_image_of_mem,
simpa only [dist_eq_norm, mem_closed_ball, mem_closed_ball_zero_iff] using zr },
{ rw [mem_closed_ball_iff_norm, add_sub_cancel],
calc ‖f z - f x - A (z - x)‖
≤ δ * ‖z - x‖ : hf _ zs _ xs
... ≤ ε * r :
mul_le_mul (le_of_lt hδ) (mem_closed_ball_iff_norm.1 zr) (norm_nonneg _) εpos.le },
{ simp only [map_sub, pi.sub_apply],
abel } },
have : A '' (closed_ball 0 r) + closed_ball (f x) (ε * r)
= {f x} + r • (A '' (closed_ball 0 1) + closed_ball 0 ε),
by rw [smul_add, ← add_assoc, add_comm ({f x}), add_assoc, smul_closed_ball _ _ εpos.le,
smul_zero, singleton_add_closed_ball_zero, ← image_smul_set ℝ E E A,
smul_closed_ball _ _ zero_le_one, smul_zero, real.norm_eq_abs, abs_of_nonneg r0, mul_one,
mul_comm],
rw this at K,
calc μ (f '' (s ∩ closed_ball x r))
≤ μ ({f x} + r • (A '' (closed_ball 0 1) + closed_ball 0 ε)) : measure_mono K
... = ennreal.of_real (r ^ finrank ℝ E) * μ (A '' closed_ball 0 1 + closed_ball 0 ε) :
by simp only [abs_of_nonneg r0, add_haar_smul, image_add_left, abs_pow, singleton_add,
measure_preimage_add]
... ≤ ennreal.of_real (r ^ finrank ℝ E) * (m * μ (closed_ball 0 1)) :
by { rw add_comm, exact ennreal.mul_le_mul le_rfl hε.le }
... = m * μ (closed_ball x r) :
by { simp only [add_haar_closed_ball' _ _ r0], ring } },
-- covering `s` by closed balls with total measure very close to `μ s`, one deduces that the
-- measure of `f '' s` is at most `m * (μ s + a)` for any positive `a`.
have J : ∀ᶠ a in 𝓝[>] (0 : ℝ≥0∞), μ (f '' s) ≤ m * (μ s + a),
{ filter_upwards [self_mem_nhds_within] with a ha,
change 0 < a at ha,
obtain ⟨t, r, t_count, ts, rpos, st, μt⟩ : ∃ (t : set E) (r : E → ℝ), t.countable ∧ t ⊆ s
∧ (∀ (x : E), x ∈ t → 0 < r x) ∧ (s ⊆ ⋃ (x ∈ t), closed_ball x (r x))
∧ ∑' (x : ↥t), μ (closed_ball ↑x (r ↑x)) ≤ μ s + a :=
besicovitch.exists_closed_ball_covering_tsum_measure_le μ ha.ne' (λ x, Ioi 0) s
(λ x xs δ δpos, ⟨δ/2, by simp [half_pos δpos, half_lt_self δpos]⟩),
haveI : encodable t := t_count.to_encodable,
calc μ (f '' s)
≤ μ (⋃ (x : t), f '' (s ∩ closed_ball x (r x))) :
begin
rw bUnion_eq_Union at st,
apply measure_mono,
rw [← image_Union, ← inter_Union],
exact image_subset _ (subset_inter (subset.refl _) st)
end
... ≤ ∑' (x : t), μ (f '' (s ∩ closed_ball x (r x))) : measure_Union_le _
... ≤ ∑' (x : t), m * μ (closed_ball x (r x)) :
ennreal.tsum_le_tsum (λ x, I x (r x) (ts x.2) (rpos x x.2).le)
... ≤ m * (μ s + a) :
by { rw ennreal.tsum_mul_left, exact ennreal.mul_le_mul le_rfl μt } },
-- taking the limit in `a`, one obtains the conclusion
have L : tendsto (λ a, (m : ℝ≥0∞) * (μ s + a)) (𝓝[>] 0) (𝓝 (m * (μ s + 0))),
{ apply tendsto.mono_left _ nhds_within_le_nhds,
apply ennreal.tendsto.const_mul (tendsto_const_nhds.add tendsto_id),
simp only [ennreal.coe_ne_top, ne.def, or_true, not_false_iff] },
rw add_zero at L,
exact ge_of_tendsto L J,
end
/-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at least `m` for any `m < det A`. -/
lemma mul_le_add_haar_image_of_lt_det
(A : E →L[ℝ] E) {m : ℝ≥0} (hm : (m : ℝ≥0∞) < ennreal.of_real (|A.det|)) :
∀ᶠ δ in 𝓝[>] (0 : ℝ≥0), ∀ (s : set E) (f : E → E) (hf : approximates_linear_on f A s δ),
(m : ℝ≥0∞) * μ s ≤ μ (f '' s) :=
begin
apply nhds_within_le_nhds,
-- The assumption `hm` implies that `A` is invertible. If `f` is close enough to `A`, it is also
-- invertible. One can then pass to the inverses, and deduce the estimate from
-- `add_haar_image_le_mul_of_det_lt` applied to `f⁻¹` and `A⁻¹`.
-- exclude first the trivial case where `m = 0`.
rcases eq_or_lt_of_le (zero_le m) with rfl|mpos,
{ apply eventually_of_forall,
simp only [forall_const, zero_mul, implies_true_iff, zero_le, ennreal.coe_zero] },
have hA : A.det ≠ 0,
{ assume h, simpa only [h, ennreal.not_lt_zero, ennreal.of_real_zero, abs_zero] using hm },
-- let `B` be the continuous linear equiv version of `A`.
let B := A.to_continuous_linear_equiv_of_det_ne_zero hA,
-- the determinant of `B.symm` is bounded by `m⁻¹`
have I : ennreal.of_real (|(B.symm : E →L[ℝ] E).det|) < (m⁻¹ : ℝ≥0),
{ simp only [ennreal.of_real, abs_inv, real.to_nnreal_inv, continuous_linear_equiv.det_coe_symm,
continuous_linear_map.coe_to_continuous_linear_equiv_of_det_ne_zero, ennreal.coe_lt_coe]
at ⊢ hm,
exact nnreal.inv_lt_inv mpos.ne' hm },
-- therefore, we may apply `add_haar_image_le_mul_of_det_lt` to `B.symm` and `m⁻¹`.
obtain ⟨δ₀, δ₀pos, hδ₀⟩ : ∃ (δ : ℝ≥0), 0 < δ ∧ ∀ (t : set E) (g : E → E),
approximates_linear_on g (B.symm : E →L[ℝ] E) t δ → μ (g '' t) ≤ ↑m⁻¹ * μ t,
{ have : ∀ᶠ (δ : ℝ≥0) in 𝓝[>] 0, ∀ (t : set E) (g : E → E),
approximates_linear_on g (B.symm : E →L[ℝ] E) t δ → μ (g '' t) ≤ ↑m⁻¹ * μ t :=
add_haar_image_le_mul_of_det_lt μ B.symm I,
rcases (this.and self_mem_nhds_within).exists with ⟨δ₀, h, h'⟩,
exact ⟨δ₀, h', h⟩, },
-- record smallness conditions for `δ` that will be needed to apply `hδ₀` below.
have L1 : ∀ᶠ δ in 𝓝 (0 : ℝ≥0), subsingleton E ∨ δ < ‖(B.symm : E →L[ℝ] E)‖₊⁻¹,
{ by_cases (subsingleton E),
{ simp only [h, true_or, eventually_const] },
simp only [h, false_or],
apply Iio_mem_nhds,
simpa only [h, false_or, inv_pos] using B.subsingleton_or_nnnorm_symm_pos },
have L2 : ∀ᶠ δ in 𝓝 (0 : ℝ≥0),
‖(B.symm : E →L[ℝ] E)‖₊ * (‖(B.symm : E →L[ℝ] E)‖₊⁻¹ - δ)⁻¹ * δ < δ₀,
{ have : tendsto (λ δ, ‖(B.symm : E →L[ℝ] E)‖₊ * (‖(B.symm : E →L[ℝ] E)‖₊⁻¹ - δ)⁻¹ * δ)
(𝓝 0) (𝓝 (‖(B.symm : E →L[ℝ] E)‖₊ * (‖(B.symm : E →L[ℝ] E)‖₊⁻¹ - 0)⁻¹ * 0)),
{ rcases eq_or_ne (‖(B.symm : E →L[ℝ] E)‖₊) 0 with H|H,
{ simpa only [H, zero_mul] using tendsto_const_nhds },
refine tendsto.mul (tendsto_const_nhds.mul _) tendsto_id,
refine (tendsto.sub tendsto_const_nhds tendsto_id).inv₀ _,
simpa only [tsub_zero, inv_eq_zero, ne.def] using H },
simp only [mul_zero] at this,
exact (tendsto_order.1 this).2 δ₀ δ₀pos },
-- let `δ` be small enough, and `f` approximated by `B` up to `δ`.
filter_upwards [L1, L2],
assume δ h1δ h2δ s f hf,
have hf' : approximates_linear_on f (B : E →L[ℝ] E) s δ,
by { convert hf, exact A.coe_to_continuous_linear_equiv_of_det_ne_zero _ },
let F := hf'.to_local_equiv h1δ,
-- the condition to be checked can be reformulated in terms of the inverse maps
suffices H : μ ((F.symm) '' F.target) ≤ (m⁻¹ : ℝ≥0) * μ F.target,
{ change (m : ℝ≥0∞) * μ (F.source) ≤ μ (F.target),
rwa [← F.symm_image_target_eq_source, mul_comm, ← ennreal.le_div_iff_mul_le, div_eq_mul_inv,
mul_comm, ← ennreal.coe_inv (mpos.ne')],
{ apply or.inl,
simpa only [ennreal.coe_eq_zero, ne.def] using mpos.ne'},
{ simp only [ennreal.coe_ne_top, true_or, ne.def, not_false_iff] } },
-- as `f⁻¹` is well approximated by `B⁻¹`, the conclusion follows from `hδ₀`
-- and our choice of `δ`.
exact hδ₀ _ _ ((hf'.to_inv h1δ).mono_num h2δ.le),
end
/-- If a differentiable function `f` is approximated by a linear map `A` on a set `s`, up to `δ`,
then at almost every `x` in `s` one has `‖f' x - A‖ ≤ δ`. -/
lemma _root_.approximates_linear_on.norm_fderiv_sub_le
{A : E →L[ℝ] E} {δ : ℝ≥0}
(hf : approximates_linear_on f A s δ) (hs : measurable_set s)
(f' : E → E →L[ℝ] E) (hf' : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) :
∀ᵐ x ∂(μ.restrict s), ‖f' x - A‖₊ ≤ δ :=
begin
/- The conclusion will hold at the Lebesgue density points of `s` (which have full measure).
At such a point `x`, for any `z` and any `ε > 0` one has for small `r`
that `{x} + r • closed_ball z ε` intersects `s`. At a point `y` in the intersection,
`f y - f x` is close both to `f' x (r z)` (by differentiability) and to `A (r z)`
(by linear approximation), so these two quantities are close, i.e., `(f' x - A) z` is small. -/
filter_upwards [besicovitch.ae_tendsto_measure_inter_div μ s, ae_restrict_mem hs],
-- start from a Lebesgue density point `x`, belonging to `s`.
assume x hx xs,
-- consider an arbitrary vector `z`.
apply continuous_linear_map.op_norm_le_bound _ δ.2 (λ z, _),
-- to show that `‖(f' x - A) z‖ ≤ δ ‖z‖`, it suffices to do it up to some error that vanishes
-- asymptotically in terms of `ε > 0`.
suffices H : ∀ ε, 0 < ε → ‖(f' x - A) z‖ ≤ (δ + ε) * (‖z‖ + ε) + ‖(f' x - A)‖ * ε,
{ have : tendsto (λ (ε : ℝ), ((δ : ℝ) + ε) * (‖z‖ + ε) + ‖(f' x - A)‖ * ε) (𝓝[>] 0)
(𝓝 ((δ + 0) * (‖z‖ + 0) + ‖(f' x - A)‖ * 0)) :=
tendsto.mono_left (continuous.tendsto (by continuity) 0) nhds_within_le_nhds,
simp only [add_zero, mul_zero] at this,
apply le_of_tendsto_of_tendsto tendsto_const_nhds this,
filter_upwards [self_mem_nhds_within],
exact H },
-- fix a positive `ε`.
assume ε εpos,
-- for small enough `r`, the rescaled ball `r • closed_ball z ε` intersects `s`, as `x` is a
-- density point
have B₁ : ∀ᶠ r in 𝓝[>] (0 : ℝ), (s ∩ ({x} + r • closed_ball z ε)).nonempty :=
eventually_nonempty_inter_smul_of_density_one μ s x hx
_ measurable_set_closed_ball (measure_closed_ball_pos μ z εpos).ne',
obtain ⟨ρ, ρpos, hρ⟩ :
∃ ρ > 0, ball x ρ ∩ s ⊆ {y : E | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖} :=
mem_nhds_within_iff.1 (is_o.def (hf' x xs) εpos),
-- for small enough `r`, the rescaled ball `r • closed_ball z ε` is included in the set where
-- `f y - f x` is well approximated by `f' x (y - x)`.
have B₂ : ∀ᶠ r in 𝓝[>] (0 : ℝ), {x} + r • closed_ball z ε ⊆ ball x ρ := nhds_within_le_nhds
(eventually_singleton_add_smul_subset bounded_closed_ball (ball_mem_nhds x ρpos)),
-- fix a small positive `r` satisfying the above properties, as well as a corresponding `y`.
obtain ⟨r, ⟨y, ⟨ys, hy⟩⟩, rρ, rpos⟩ : ∃ (r : ℝ), (s ∩ ({x} + r • closed_ball z ε)).nonempty ∧
{x} + r • closed_ball z ε ⊆ ball x ρ ∧ 0 < r := (B₁.and (B₂.and self_mem_nhds_within)).exists,
-- write `y = x + r a` with `a ∈ closed_ball z ε`.
obtain ⟨a, az, ya⟩ : ∃ a, a ∈ closed_ball z ε ∧ y = x + r • a,
{ simp only [mem_smul_set, image_add_left, mem_preimage, singleton_add] at hy,
rcases hy with ⟨a, az, ha⟩,
exact ⟨a, az, by simp only [ha, add_neg_cancel_left]⟩ },
have norm_a : ‖a‖ ≤ ‖z‖ + ε := calc
‖a‖ = ‖z + (a - z)‖ : by simp only [add_sub_cancel'_right]
... ≤ ‖z‖ + ‖a - z‖ : norm_add_le _ _
... ≤ ‖z‖ + ε : add_le_add_left (mem_closed_ball_iff_norm.1 az) _,
-- use the approximation properties to control `(f' x - A) a`, and then `(f' x - A) z` as `z` is
-- close to `a`.
have I : r * ‖(f' x - A) a‖ ≤ r * (δ + ε) * (‖z‖ + ε) := calc
r * ‖(f' x - A) a‖ = ‖(f' x - A) (r • a)‖ :
by simp only [continuous_linear_map.map_smul, norm_smul, real.norm_eq_abs,
abs_of_nonneg rpos.le]
... = ‖(f y - f x - A (y - x)) -
(f y - f x - (f' x) (y - x))‖ :
begin
congr' 1,
simp only [ya, add_sub_cancel', sub_sub_sub_cancel_left, continuous_linear_map.coe_sub',
eq_self_iff_true, sub_left_inj, pi.sub_apply, continuous_linear_map.map_smul, smul_sub],
end
... ≤ ‖f y - f x - A (y - x)‖ +
‖f y - f x - (f' x) (y - x)‖ : norm_sub_le _ _
... ≤ δ * ‖y - x‖ + ε * ‖y - x‖ :
add_le_add (hf _ ys _ xs) (hρ ⟨rρ hy, ys⟩)
... = r * (δ + ε) * ‖a‖ :
by { simp only [ya, add_sub_cancel', norm_smul, real.norm_eq_abs, abs_of_nonneg rpos.le],
ring }
... ≤ r * (δ + ε) * (‖z‖ + ε) :
mul_le_mul_of_nonneg_left norm_a (mul_nonneg rpos.le (add_nonneg δ.2 εpos.le)),
show ‖(f' x - A) z‖ ≤ (δ + ε) * (‖z‖ + ε) + ‖(f' x - A)‖ * ε, from calc
‖(f' x - A) z‖ = ‖(f' x - A) a + (f' x - A) (z - a)‖ :
begin
congr' 1,
simp only [continuous_linear_map.coe_sub', map_sub, pi.sub_apply],
abel
end
... ≤ ‖(f' x - A) a‖ + ‖(f' x - A) (z - a)‖ : norm_add_le _ _
... ≤ (δ + ε) * (‖z‖ + ε) + ‖f' x - A‖ * ‖z - a‖ :
begin
apply add_le_add,
{ rw mul_assoc at I, exact (mul_le_mul_left rpos).1 I },
{ apply continuous_linear_map.le_op_norm }
end
... ≤ (δ + ε) * (‖z‖ + ε) + ‖f' x - A‖ * ε : add_le_add le_rfl
(mul_le_mul_of_nonneg_left (mem_closed_ball_iff_norm'.1 az) (norm_nonneg _)),
end
/-!
### Measure zero of the image, over non-measurable sets
If a set has measure `0`, then its image under a differentiable map has measure zero. This doesn't
require the set to be measurable. In the same way, if `f` is differentiable on a set `s` with
non-invertible derivative everywhere, then `f '' s` has measure `0`, again without measurability
assumptions.
-/
/-- A differentiable function maps sets of measure zero to sets of measure zero. -/
lemma add_haar_image_eq_zero_of_differentiable_on_of_add_haar_eq_zero
(hf : differentiable_on ℝ f s) (hs : μ s = 0) :
μ (f '' s) = 0 :=
begin
refine le_antisymm _ (zero_le _),
have : ∀ (A : E →L[ℝ] E), ∃ (δ : ℝ≥0), 0 < δ ∧ ∀ (t : set E)
(hf : approximates_linear_on f A t δ), μ (f '' t) ≤ (real.to_nnreal (|A.det|) + 1 : ℝ≥0) * μ t,
{ assume A,
let m : ℝ≥0 := real.to_nnreal ((|A.det|)) + 1,
have I : ennreal.of_real (|A.det|) < m,
by simp only [ennreal.of_real, m, lt_add_iff_pos_right, zero_lt_one, ennreal.coe_lt_coe],
rcases ((add_haar_image_le_mul_of_det_lt μ A I).and self_mem_nhds_within).exists
with ⟨δ, h, h'⟩,
exact ⟨δ, h', λ t ht, h t f ht⟩ },
choose δ hδ using this,
obtain ⟨t, A, t_disj, t_meas, t_cover, ht, -⟩ : ∃ (t : ℕ → set E) (A : ℕ → (E →L[ℝ] E)),
pairwise (disjoint on t) ∧ (∀ (n : ℕ), measurable_set (t n)) ∧ (s ⊆ ⋃ (n : ℕ), t n)
∧ (∀ (n : ℕ), approximates_linear_on f (A n) (s ∩ t n) (δ (A n)))
∧ (s.nonempty → ∀ n, ∃ y ∈ s, A n = fderiv_within ℝ f s y) :=
exists_partition_approximates_linear_on_of_has_fderiv_within_at f s
(fderiv_within ℝ f s) (λ x xs, (hf x xs).has_fderiv_within_at) δ (λ A, (hδ A).1.ne'),
calc μ (f '' s)
≤ μ (⋃ n, f '' (s ∩ t n)) :
begin
apply measure_mono,
rw [← image_Union, ← inter_Union],
exact image_subset f (subset_inter subset.rfl t_cover)
end
... ≤ ∑' n, μ (f '' (s ∩ t n)) : measure_Union_le _
... ≤ ∑' n, (real.to_nnreal (|(A n).det|) + 1 : ℝ≥0) * μ (s ∩ t n) :
begin
apply ennreal.tsum_le_tsum (λ n, _),
apply (hδ (A n)).2,
exact ht n,
end
... ≤ ∑' n, (real.to_nnreal (|(A n).det|) + 1 : ℝ≥0) * 0 :
begin
refine ennreal.tsum_le_tsum (λ n, ennreal.mul_le_mul le_rfl _),
exact le_trans (measure_mono (inter_subset_left _ _)) (le_of_eq hs),
end
... = 0 : by simp only [tsum_zero, mul_zero]
end
/-- A version of Sard lemma in fixed dimension: given a differentiable function from `E` to `E` and
a set where the differential is not invertible, then the image of this set has zero measure.
Here, we give an auxiliary statement towards this result. -/
lemma add_haar_image_eq_zero_of_det_fderiv_within_eq_zero_aux
(hf' : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x)
(R : ℝ) (hs : s ⊆ closed_ball 0 R) (ε : ℝ≥0) (εpos : 0 < ε)
(h'f' : ∀ x ∈ s, (f' x).det = 0) :
μ (f '' s) ≤ ε * μ (closed_ball 0 R) :=
begin
rcases eq_empty_or_nonempty s with rfl|h's, { simp only [measure_empty, zero_le, image_empty] },
have : ∀ (A : E →L[ℝ] E), ∃ (δ : ℝ≥0), 0 < δ ∧ ∀ (t : set E)
(hf : approximates_linear_on f A t δ), μ (f '' t) ≤ (real.to_nnreal (|A.det|) + ε : ℝ≥0) * μ t,
{ assume A,
let m : ℝ≥0 := real.to_nnreal (|A.det|) + ε,
have I : ennreal.of_real (|A.det|) < m,
by simp only [ennreal.of_real, m, lt_add_iff_pos_right, εpos, ennreal.coe_lt_coe],
rcases ((add_haar_image_le_mul_of_det_lt μ A I).and self_mem_nhds_within).exists
with ⟨δ, h, h'⟩,
exact ⟨δ, h', λ t ht, h t f ht⟩ },
choose δ hδ using this,
obtain ⟨t, A, t_disj, t_meas, t_cover, ht, Af'⟩ : ∃ (t : ℕ → set E) (A : ℕ → (E →L[ℝ] E)),
pairwise (disjoint on t) ∧ (∀ (n : ℕ), measurable_set (t n)) ∧ (s ⊆ ⋃ (n : ℕ), t n)
∧ (∀ (n : ℕ), approximates_linear_on f (A n) (s ∩ t n) (δ (A n)))
∧ (s.nonempty → ∀ n, ∃ y ∈ s, A n = f' y) :=
exists_partition_approximates_linear_on_of_has_fderiv_within_at f s
f' hf' δ (λ A, (hδ A).1.ne'),
calc μ (f '' s)
≤ μ (⋃ n, f '' (s ∩ t n)) :
begin
apply measure_mono,
rw [← image_Union, ← inter_Union],
exact image_subset f (subset_inter subset.rfl t_cover)
end
... ≤ ∑' n, μ (f '' (s ∩ t n)) : measure_Union_le _
... ≤ ∑' n, (real.to_nnreal (|(A n).det|) + ε : ℝ≥0) * μ (s ∩ t n) :
begin
apply ennreal.tsum_le_tsum (λ n, _),
apply (hδ (A n)).2,
exact ht n,
end
... = ∑' n, ε * μ (s ∩ t n) :
begin
congr' with n,
rcases Af' h's n with ⟨y, ys, hy⟩,
simp only [hy, h'f' y ys, real.to_nnreal_zero, abs_zero, zero_add]
end
... ≤ ε * ∑' n, μ (closed_ball 0 R ∩ t n) :
begin
rw ennreal.tsum_mul_left,
refine ennreal.mul_le_mul le_rfl (ennreal.tsum_le_tsum (λ n, measure_mono _)),
exact inter_subset_inter_left _ hs,
end
... = ε * μ (⋃ n, closed_ball 0 R ∩ t n) :
begin
rw measure_Union,
{ exact pairwise_disjoint.mono t_disj (λ n, inter_subset_right _ _) },
{ assume n,
exact measurable_set_closed_ball.inter (t_meas n) }
end
... ≤ ε * μ (closed_ball 0 R) :
begin
rw ← inter_Union,
exact ennreal.mul_le_mul le_rfl (measure_mono (inter_subset_left _ _)),
end
end
/-- A version of Sard lemma in fixed dimension: given a differentiable function from `E` to `E` and
a set where the differential is not invertible, then the image of this set has zero measure. -/
lemma add_haar_image_eq_zero_of_det_fderiv_within_eq_zero
(hf' : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x)
(h'f' : ∀ x ∈ s, (f' x).det = 0) :
μ (f '' s) = 0 :=
begin
suffices H : ∀ R, μ (f '' (s ∩ closed_ball 0 R)) = 0,
{ apply le_antisymm _ (zero_le _),
rw ← Union_inter_closed_ball_nat s 0,
calc μ (f '' ⋃ (n : ℕ), s ∩ closed_ball 0 n) ≤ ∑' (n : ℕ), μ (f '' (s ∩ closed_ball 0 n)) :
by { rw image_Union, exact measure_Union_le _ }
... ≤ 0 : by simp only [H, tsum_zero, nonpos_iff_eq_zero] },
assume R,
have A : ∀ (ε : ℝ≥0) (εpos : 0 < ε), μ (f '' (s ∩ closed_ball 0 R)) ≤ ε * μ (closed_ball 0 R) :=
λ ε εpos, add_haar_image_eq_zero_of_det_fderiv_within_eq_zero_aux μ
(λ x hx, (hf' x hx.1).mono (inter_subset_left _ _)) R (inter_subset_right _ _) ε εpos
(λ x hx, h'f' x hx.1),
have B : tendsto (λ (ε : ℝ≥0), (ε : ℝ≥0∞) * μ (closed_ball 0 R)) (𝓝[>] 0) (𝓝 0),
{ have : tendsto (λ (ε : ℝ≥0), (ε : ℝ≥0∞) * μ (closed_ball 0 R))
(𝓝 0) (𝓝 (((0 : ℝ≥0) : ℝ≥0∞) * μ (closed_ball 0 R))) :=
ennreal.tendsto.mul_const (ennreal.tendsto_coe.2 tendsto_id)
(or.inr ((measure_closed_ball_lt_top).ne)),
simp only [zero_mul, ennreal.coe_zero] at this,
exact tendsto.mono_left this nhds_within_le_nhds },
apply le_antisymm _ (zero_le _),
apply ge_of_tendsto B,
filter_upwards [self_mem_nhds_within],
exact A,
end
/-!
### Weak measurability statements
We show that the derivative of a function on a set is almost everywhere measurable, and that the
image `f '' s` is measurable if `f` is injective on `s`. The latter statement follows from the
Lusin-Souslin theorem.
-/
/-- The derivative of a function on a measurable set is almost everywhere measurable on this set
with respect to Lebesgue measure. Note that, in general, it is not genuinely measurable there,
as `f'` is not unique (but only on a set of measure `0`, as the argument shows). -/
lemma ae_measurable_fderiv_within (hs : measurable_set s)
(hf' : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) :
ae_measurable f' (μ.restrict s) :=
begin
/- It suffices to show that `f'` can be uniformly approximated by a measurable function.
Fix `ε > 0`. Thanks to `exists_partition_approximates_linear_on_of_has_fderiv_within_at`, one
can find a countable measurable partition of `s` into sets `s ∩ t n` on which `f` is well
approximated by linear maps `A n`. On almost all of `s ∩ t n`, it follows from
`approximates_linear_on.norm_fderiv_sub_le` that `f'` is uniformly approximated by `A n`, which
gives the conclusion. -/
-- fix a precision `ε`
refine ae_measurable_of_unif_approx (λ ε εpos, _),
let δ : ℝ≥0 := ⟨ε, le_of_lt εpos⟩,
have δpos : 0 < δ := εpos,
-- partition `s` into sets `s ∩ t n` on which `f` is approximated by linear maps `A n`.
obtain ⟨t, A, t_disj, t_meas, t_cover, ht, Af'⟩ : ∃ (t : ℕ → set E) (A : ℕ → (E →L[ℝ] E)),
pairwise (disjoint on t) ∧ (∀ (n : ℕ), measurable_set (t n)) ∧ (s ⊆ ⋃ (n : ℕ), t n)
∧ (∀ (n : ℕ), approximates_linear_on f (A n) (s ∩ t n) δ)
∧ (s.nonempty → ∀ n, ∃ y ∈ s, A n = f' y) :=
exists_partition_approximates_linear_on_of_has_fderiv_within_at f s
f' hf' (λ A, δ) (λ A, δpos.ne'),
-- define a measurable function `g` which coincides with `A n` on `t n`.
obtain ⟨g, g_meas, hg⟩ : ∃ g : E → (E →L[ℝ] E), measurable g ∧
∀ (n : ℕ) (x : E), x ∈ t n → g x = A n :=
exists_measurable_piecewise_nat t t_meas t_disj (λ n x, A n) (λ n, measurable_const),
refine ⟨g, g_meas.ae_measurable, _⟩,
-- reduce to checking that `f'` and `g` are close on almost all of `s ∩ t n`, for all `n`.
suffices H : ∀ᵐ (x : E) ∂(sum (λ n, μ.restrict (s ∩ t n))), dist (g x) (f' x) ≤ ε,
{ have : μ.restrict s ≤ sum (λ n, μ.restrict (s ∩ t n)),
{ have : s = ⋃ n, s ∩ t n,
{ rw ← inter_Union,
exact subset.antisymm (subset_inter subset.rfl t_cover) (inter_subset_left _ _) },
conv_lhs { rw this },
exact restrict_Union_le },
exact ae_mono this H },
-- fix such an `n`.
refine ae_sum_iff.2 (λ n, _),
-- on almost all `s ∩ t n`, `f' x` is close to `A n` thanks to
-- `approximates_linear_on.norm_fderiv_sub_le`.
have E₁ : ∀ᵐ (x : E) ∂μ.restrict (s ∩ t n), ‖f' x - A n‖₊ ≤ δ :=
(ht n).norm_fderiv_sub_le μ (hs.inter (t_meas n)) f'
(λ x hx, (hf' x hx.1).mono (inter_subset_left _ _)),
-- moreover, `g x` is equal to `A n` there.
have E₂ : ∀ᵐ (x : E) ∂μ.restrict (s ∩ t n), g x = A n,
{ suffices H : ∀ᵐ (x : E) ∂μ.restrict (t n), g x = A n,
from ae_mono (restrict_mono (inter_subset_right _ _) le_rfl) H,
filter_upwards [ae_restrict_mem (t_meas n)],
exact hg n },
-- putting these two properties together gives the conclusion.
filter_upwards [E₁, E₂] with x hx1 hx2,
rw ← nndist_eq_nnnorm at hx1,
rw [hx2, dist_comm],
exact hx1,
end
lemma ae_measurable_of_real_abs_det_fderiv_within (hs : measurable_set s)
(hf' : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) :
ae_measurable (λ x, ennreal.of_real (|(f' x).det|)) (μ.restrict s) :=
begin
apply ennreal.measurable_of_real.comp_ae_measurable,
refine continuous_abs.measurable.comp_ae_measurable _,
refine continuous_linear_map.continuous_det.measurable.comp_ae_measurable _,
exact ae_measurable_fderiv_within μ hs hf'
end
lemma ae_measurable_to_nnreal_abs_det_fderiv_within (hs : measurable_set s)
(hf' : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) :
ae_measurable (λ x, |(f' x).det|.to_nnreal) (μ.restrict s) :=
begin
apply measurable_real_to_nnreal.comp_ae_measurable,
refine continuous_abs.measurable.comp_ae_measurable _,
refine continuous_linear_map.continuous_det.measurable.comp_ae_measurable _,
exact ae_measurable_fderiv_within μ hs hf'
end
/-- If a function is differentiable and injective on a measurable set,
then the image is measurable.-/
lemma measurable_image_of_fderiv_within (hs : measurable_set s)
(hf' : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) (hf : inj_on f s) :
measurable_set (f '' s) :=
begin
have : differentiable_on ℝ f s := λ x hx, (hf' x hx).differentiable_within_at,
exact hs.image_of_continuous_on_inj_on (differentiable_on.continuous_on this) hf,
end
/-- If a function is differentiable and injective on a measurable set `s`, then its restriction
to `s` is a measurable embedding. -/
lemma measurable_embedding_of_fderiv_within (hs : measurable_set s)
(hf' : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) (hf : inj_on f s) :
measurable_embedding (s.restrict f) :=
begin
have : differentiable_on ℝ f s := λ x hx, (hf' x hx).differentiable_within_at,
exact this.continuous_on.measurable_embedding hs hf
end
/-!
### Proving the estimate for the measure of the image
We show the formula `∫⁻ x in s, ennreal.of_real (|(f' x).det|) ∂μ = μ (f '' s)`,
in `lintegral_abs_det_fderiv_eq_add_haar_image`. For this, we show both inequalities in both
directions, first up to controlled errors and then letting these errors tend to `0`.
-/
lemma add_haar_image_le_lintegral_abs_det_fderiv_aux1 (hs : measurable_set s)
(hf' : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) {ε : ℝ≥0} (εpos : 0 < ε) :
μ (f '' s) ≤ ∫⁻ x in s, ennreal.of_real (|(f' x).det|) ∂μ + 2 * ε * μ s :=
begin
/- To bound `μ (f '' s)`, we cover `s` by sets where `f` is well-approximated by linear maps
`A n` (and where `f'` is almost everywhere close to `A n`), and then use that `f` expands the
measure of such a set by at most `(A n).det + ε`. -/
have : ∀ (A : E →L[ℝ] E), ∃ (δ : ℝ≥0), 0 < δ ∧
(∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ δ → |B.det - A.det| ≤ ε) ∧
∀ (t : set E) (g : E → E) (hf : approximates_linear_on g A t δ),
μ (g '' t) ≤ (ennreal.of_real (|A.det|) + ε) * μ t,
{ assume A,
let m : ℝ≥0 := real.to_nnreal (|A.det|) + ε,
have I : ennreal.of_real (|A.det|) < m,
by simp only [ennreal.of_real, m, lt_add_iff_pos_right, εpos, ennreal.coe_lt_coe],
rcases ((add_haar_image_le_mul_of_det_lt μ A I).and self_mem_nhds_within).exists
with ⟨δ, h, δpos⟩,
obtain ⟨δ', δ'pos, hδ'⟩ :
∃ (δ' : ℝ) (H : 0 < δ'), ∀ B, dist B A < δ' → dist B.det A.det < ↑ε :=
continuous_at_iff.1 continuous_linear_map.continuous_det.continuous_at ε εpos,
let δ'' : ℝ≥0 := ⟨δ' / 2, (half_pos δ'pos).le⟩,
refine ⟨min δ δ'', lt_min δpos (half_pos δ'pos), _, _⟩,
{ assume B hB,
rw ← real.dist_eq,
apply (hδ' B _).le,
rw dist_eq_norm,
calc ‖B - A‖ ≤ (min δ δ'' : ℝ≥0) : hB
... ≤ δ'' : by simp only [le_refl, nnreal.coe_min, min_le_iff, or_true]
... < δ' : half_lt_self δ'pos },
{ assume t g htg,
exact h t g (htg.mono_num (min_le_left _ _)) } },
choose δ hδ using this,
obtain ⟨t, A, t_disj, t_meas, t_cover, ht, -⟩ : ∃ (t : ℕ → set E) (A : ℕ → (E →L[ℝ] E)),
pairwise (disjoint on t) ∧ (∀ (n : ℕ), measurable_set (t n)) ∧ (s ⊆ ⋃ (n : ℕ), t n)
∧ (∀ (n : ℕ), approximates_linear_on f (A n) (s ∩ t n) (δ (A n)))
∧ (s.nonempty → ∀ n, ∃ y ∈ s, A n = f' y) :=
exists_partition_approximates_linear_on_of_has_fderiv_within_at f s
f' hf' δ (λ A, (hδ A).1.ne'),
calc μ (f '' s)
≤ μ (⋃ n, f '' (s ∩ t n)) :
begin
apply measure_mono,
rw [← image_Union, ← inter_Union],
exact image_subset f (subset_inter subset.rfl t_cover)
end
... ≤ ∑' n, μ (f '' (s ∩ t n)) : measure_Union_le _
... ≤ ∑' n, (ennreal.of_real (|(A n).det|) + ε) * μ (s ∩ t n) :
begin
apply ennreal.tsum_le_tsum (λ n, _),
apply (hδ (A n)).2.2,
exact ht n,
end
... = ∑' n, ∫⁻ x in s ∩ t n, ennreal.of_real (|(A n).det|) + ε ∂μ :
by simp only [lintegral_const, measurable_set.univ, measure.restrict_apply, univ_inter]
... ≤ ∑' n, ∫⁻ x in s ∩ t n, ennreal.of_real (|(f' x).det|) + 2 * ε ∂μ :
begin
apply ennreal.tsum_le_tsum (λ n, _),
apply lintegral_mono_ae,
filter_upwards [(ht n).norm_fderiv_sub_le μ (hs.inter (t_meas n)) f'
(λ x hx, (hf' x hx.1).mono (inter_subset_left _ _))],
assume x hx,
have I : |(A n).det| ≤ |(f' x).det| + ε := calc
|(A n).det| = |(f' x).det - ((f' x).det - (A n).det)| : by { congr' 1, abel }
... ≤ |(f' x).det| + |(f' x).det - (A n).det| : abs_sub _ _
... ≤ |(f' x).det| + ε : add_le_add le_rfl ((hδ (A n)).2.1 _ hx),
calc ennreal.of_real (|(A n).det|) + ε
≤ ennreal.of_real (|(f' x).det| + ε) + ε :
add_le_add (ennreal.of_real_le_of_real I) le_rfl
... = ennreal.of_real (|(f' x).det|) + 2 * ε :
by simp only [ennreal.of_real_add, abs_nonneg, two_mul, add_assoc, nnreal.zero_le_coe,
ennreal.of_real_coe_nnreal],
end
... = ∫⁻ x in ⋃ n, s ∩ t n, ennreal.of_real (|(f' x).det|) + 2 * ε ∂μ :
begin
have M : ∀ (n : ℕ), measurable_set (s ∩ t n) := λ n, hs.inter (t_meas n),
rw lintegral_Union M,
exact pairwise_disjoint.mono t_disj (λ n, inter_subset_right _ _),
end
... = ∫⁻ x in s, ennreal.of_real (|(f' x).det|) + 2 * ε ∂μ :
begin
have : s = ⋃ n, s ∩ t n,
{ rw ← inter_Union,
exact subset.antisymm (subset_inter subset.rfl t_cover) (inter_subset_left _ _) },
rw ← this,
end
... = ∫⁻ x in s, ennreal.of_real (|(f' x).det|) ∂μ + 2 * ε * μ s :
by simp only [lintegral_add_right' _ ae_measurable_const, set_lintegral_const]
end
lemma add_haar_image_le_lintegral_abs_det_fderiv_aux2 (hs : measurable_set s) (h's : μ s ≠ ∞)
(hf' : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) :
μ (f '' s) ≤ ∫⁻ x in s, ennreal.of_real (|(f' x).det|) ∂μ :=
begin
/- We just need to let the error tend to `0` in the previous lemma. -/
have : tendsto (λ (ε : ℝ≥0), ∫⁻ x in s, ennreal.of_real (|(f' x).det|) ∂μ + 2 * ε * μ s)
(𝓝[>] 0) (𝓝 (∫⁻ x in s, ennreal.of_real (|(f' x).det|) ∂μ + 2 * (0 : ℝ≥0) * μ s)),
{ apply tendsto.mono_left _ nhds_within_le_nhds,
refine tendsto_const_nhds.add _,
refine ennreal.tendsto.mul_const _ (or.inr h's),
exact ennreal.tendsto.const_mul (ennreal.tendsto_coe.2 tendsto_id)
(or.inr ennreal.coe_ne_top) },
simp only [add_zero, zero_mul, mul_zero, ennreal.coe_zero] at this,
apply ge_of_tendsto this,
filter_upwards [self_mem_nhds_within],
rintros ε (εpos : 0 < ε),
exact add_haar_image_le_lintegral_abs_det_fderiv_aux1 μ hs hf' εpos,
end
lemma add_haar_image_le_lintegral_abs_det_fderiv (hs : measurable_set s)
(hf' : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) :
μ (f '' s) ≤ ∫⁻ x in s, ennreal.of_real (|(f' x).det|) ∂μ :=
begin
/- We already know the result for finite-measure sets. We cover `s` by finite-measure sets using
`spanning_sets μ`, and apply the previous result to each of these parts. -/
let u := λ n, disjointed (spanning_sets μ) n,
have u_meas : ∀ n, measurable_set (u n),
{ assume n,
apply measurable_set.disjointed (λ i, _),
exact measurable_spanning_sets μ i },
have A : s = ⋃ n, s ∩ u n,
by rw [← inter_Union, Union_disjointed, Union_spanning_sets, inter_univ],
calc μ (f '' s) ≤ ∑' n, μ (f '' (s ∩ u n)) :
begin
conv_lhs { rw [A, image_Union] },
exact measure_Union_le _,
end
... ≤ ∑' n, ∫⁻ x in s ∩ u n, ennreal.of_real (|(f' x).det|) ∂μ :
begin
apply ennreal.tsum_le_tsum (λ n, _),
apply add_haar_image_le_lintegral_abs_det_fderiv_aux2 μ (hs.inter (u_meas n)) _
(λ x hx, (hf' x hx.1).mono (inter_subset_left _ _)),
have : μ (u n) < ∞ :=
lt_of_le_of_lt (measure_mono (disjointed_subset _ _)) (measure_spanning_sets_lt_top μ n),
exact ne_of_lt (lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) this),
end
... = ∫⁻ x in s, ennreal.of_real (|(f' x).det|) ∂μ :
begin
conv_rhs { rw A },
rw lintegral_Union,
{ assume n, exact hs.inter (u_meas n) },
{ exact pairwise_disjoint.mono (disjoint_disjointed _) (λ n, inter_subset_right _ _) }
end
end
lemma lintegral_abs_det_fderiv_le_add_haar_image_aux1 (hs : measurable_set s)
(hf' : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) (hf : inj_on f s)
{ε : ℝ≥0} (εpos : 0 < ε) :
∫⁻ x in s, ennreal.of_real (|(f' x).det|) ∂μ ≤ μ (f '' s) + 2 * ε * μ s :=
begin
/- To bound `∫⁻ x in s, ennreal.of_real (|(f' x).det|) ∂μ`, we cover `s` by sets where `f` is
well-approximated by linear maps `A n` (and where `f'` is almost everywhere close to `A n`),
and then use that `f` expands the measure of such a set by at least `(A n).det - ε`. -/
have : ∀ (A : E →L[ℝ] E), ∃ (δ : ℝ≥0), 0 < δ ∧
(∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ δ → |B.det - A.det| ≤ ε) ∧
∀ (t : set E) (g : E → E) (hf : approximates_linear_on g A t δ),
ennreal.of_real (|A.det|) * μ t ≤ μ (g '' t) + ε * μ t,
{ assume A,
obtain ⟨δ', δ'pos, hδ'⟩ :
∃ (δ' : ℝ) (H : 0 < δ'), ∀ B, dist B A < δ' → dist B.det A.det < ↑ε :=
continuous_at_iff.1 continuous_linear_map.continuous_det.continuous_at ε εpos,
let δ'' : ℝ≥0 := ⟨δ' / 2, (half_pos δ'pos).le⟩,
have I'' : ∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ'' → |B.det - A.det| ≤ ↑ε,
{ assume B hB,
rw ← real.dist_eq,
apply (hδ' B _).le,
rw dist_eq_norm,
exact hB.trans_lt (half_lt_self δ'pos) },
rcases eq_or_ne A.det 0 with hA|hA,
{ refine ⟨δ'', half_pos δ'pos, I'', _⟩,
simp only [hA, forall_const, zero_mul, ennreal.of_real_zero, implies_true_iff, zero_le,
abs_zero] },
let m : ℝ≥0 := real.to_nnreal (|A.det|) - ε,
have I : (m : ℝ≥0∞) < ennreal.of_real (|A.det|),
{ simp only [ennreal.of_real, with_top.coe_sub],
apply ennreal.sub_lt_self ennreal.coe_ne_top,
{ simpa only [abs_nonpos_iff, real.to_nnreal_eq_zero, ennreal.coe_eq_zero, ne.def] using hA },
{ simp only [εpos.ne', ennreal.coe_eq_zero, ne.def, not_false_iff] } },
rcases ((mul_le_add_haar_image_of_lt_det μ A I).and self_mem_nhds_within).exists
with ⟨δ, h, δpos⟩,
refine ⟨min δ δ'', lt_min δpos (half_pos δ'pos), _, _⟩,
{ assume B hB,
apply I'' _ (hB.trans _),
simp only [le_refl, nnreal.coe_min, min_le_iff, or_true] },
{ assume t g htg,
rcases eq_or_ne (μ t) ∞ with ht|ht,
{ simp only [ht, εpos.ne', with_top.mul_top, ennreal.coe_eq_zero, le_top, ne.def,
not_false_iff, ennreal.add_top] },
have := h t g (htg.mono_num (min_le_left _ _)),
rwa [with_top.coe_sub, ennreal.sub_mul, tsub_le_iff_right] at this,
simp only [ht, implies_true_iff, ne.def, not_false_iff] } },
choose δ hδ using this,
obtain ⟨t, A, t_disj, t_meas, t_cover, ht, -⟩ : ∃ (t : ℕ → set E) (A : ℕ → (E →L[ℝ] E)),
pairwise (disjoint on t) ∧ (∀ (n : ℕ), measurable_set (t n)) ∧ (s ⊆ ⋃ (n : ℕ), t n)
∧ (∀ (n : ℕ), approximates_linear_on f (A n) (s ∩ t n) (δ (A n)))