Lebesgue decomposition of the Monge–Ampère measure #
Let E be a finite-dimensional real inner product space with an additive Haar measure μ, and
let f : E → EReal be convex (convex real epigraph, never ⊥), with D the interior of its
effective domain and F x = (f x).toReal. By Alexandrov's theorem, at almost every x ∈ D the
subdifferential of f is differentiable, with a positive derivative H x, the Alexandrov Hessian
of f at x: every subgradient y' at a point x' near x satisfies
y' = ∇F x + H x (x' - x) + o(‖x' - x‖). This file shows that the Alexandrov Hessian determines
exactly the absolutely continuous part of the Monge–Ampère measure MA_{f, g μ} weighted by a
measurable density g, MA_{f, g μ}(s) = ∫_{∂f(s ∩ D)} g dμ:
MA_{f, g μ} = (MA_{f, g μ})ˢ + g (∇F x) det (H x) dx on D,
where (MA_{f, g μ})ˢ is singular with respect to μ. For g = 1 this is the decomposition
MA_f = (MA_f)ˢ + det (H x) dx of the Aleksandrov measure. So MA_f = h dx gives det (H x) = h x
almost everywhere: an Aleksandrov solution of the Monge–Ampère equation det D²f = h solves it
almost everywhere. The converse fails, because the singular part is invisible to the Alexandrov
Hessian: a cone has Alexandrov Hessian determinant 0 almost everywhere, yet its Monge–Ampère
measure is a positive multiple of a Dirac mass (TauCeti.mongeAmpereMeasure_mul_norm_sub).
For weighted measures this is the Jacobian equation of optimal transport. If the Monge–Ampère
measure of f weighted by ν₁ = g₁ dx is ν₀ = g₀ dx, which is what it means for f to be a
weak (Brenier) solution of the transport problem from ν₀ to ν₁
(TauCeti.mongeAmpereMeasure_eq_of_hasLaw), then g₀ x = g₁ (∇F x) det (H x) for almost every
x ∈ D.
The decomposition is proved by the change of variables formula. Off a null set of D, the
subdifferential of f at x is the single gradient ∇F x, and the gradient map has derivative
H x within that set. If two such points x₁ ≠ x₂ have the same gradient y, then y is a
subgradient along the segment between them, so H x₁ (x₂ - x₁) = 0
(TauCeti.apply_sub_eq_zero_of_forall_mem_subdifferential_norm_sub_le). So the gradient is
injective where det (H x) ≠ 0, where Mathlib's change of variables formula computes the weighted
measure of its image, and it maps the set where det (H x) = 0 to a null set
(TauCeti.withDensity_image_gradient_eq_setLIntegral). The remaining part
of MA_{f, g μ} lives on a μ-null subset of D, hence is singular.
Separately, the density is identified pointwise: MA_f(closedBall x r) / μ(closedBall x r) → det A
as r → 0⁺ at every x ∈ D where the subdifferential has derivative A. Up to o(r), the
subgradients at points of closedBall x r lie in ∇F x + r • A (closedBall 0 1), which gives the
upper bound. For the lower bound, A is coercive when det A ≠ 0, so for s < 1 and small r
the second-order expansion of F shows that, for each p ∈ ∇F x + r • A (closedBall 0 s), the
convex function F - ⟪·, p⟫ is smaller somewhere in the ball than on its sphere; its minimum over
the ball is then attained inside, where p is a subgradient.
Main statements #
TauCeti.tendsto_mongeAmpereMeasure_closedBall_div— the density ofMA_fat a point where the subdifferential is differentiable with derivativeAisdet A;TauCeti.rnDeriv_mongeAmpereMeasure_withDensity_ae_eqandTauCeti.rnDeriv_mongeAmpereMeasure_ae_eq— the Radon–Nikodym derivative ofMA_{f, g μ}isg (∇F x) det (H x)onD, and0offD; forg = 1it isdet (H x);TauCeti.mongeAmpereMeasure_withDensity_eq_singularPart_add_withDensity,TauCeti.mongeAmpereMeasure_eq_singularPart_add_withDensityandTauCeti.exists_mongeAmpereMeasure_eq_singularPart_add_withDensity— the Lebesgue decompositionMA_{f, g μ} = (MA_{f, g μ})ˢ + g (∇F x) det (H x) dx, and its caseMA_f = (MA_f)ˢ + det (H x) dx;TauCeti.ae_eq_indicator_ofReal_det_of_mongeAmpereMeasure_eq_withDensity— an Aleksandrov solution ofdet D²f = gsatisfiesdet (H x) = g xalmost everywhere;TauCeti.rnDeriv_ae_eq_indicator_ofReal_det_mul_rnDeriv_of_mongeAmpereMeasure_eq— the Jacobian equation: a weak solution of the transport problem fromg₀ dxtog₁ dxsatisfiesg₀ x = g₁ (∇F x) det (H x)almost everywhere.
References #
- C. E. Gutiérrez, The Monge–Ampère Equation, 2nd ed., Progress in Nonlinear Differential Equations and Their Applications 89, Birkhäuser, 2016, §1.1, for the Monge–Ampère measure.
- A. Figalli, The Monge–Ampère Equation and Its Applications, Zurich Lectures in Advanced Mathematics, EMS, 2017, §2.1.
- L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, revised ed., CRC Press, 2015, §6.4, for Alexandrov's theorem and the splitting of the Hessian measure of a convex function into absolutely continuous and singular parts.
- R. J. McCann, A convexity principle for interacting gases, Adv. Math. 128 (1997), 153–179, Theorem 4.4, for the Jacobian equation of a transport map given by a convex gradient.
- C. Villani, Optimal Transport: Old and New, Grundlehren 338, Springer, 2009, Chapter 11.
The density of the Monge–Ampère measure at a point of second-order differentiability. Let
f : E → EReal be convex (convex real epigraph, never ⊥) and x an interior point of its
effective domain at which the subdifferential has the first-order expansion
y' = y + A (x' - x) + o(‖x' - x‖), uniformly over the subgradients y' at points x' near
x. Then for every additive Haar measure μ, the Aleksandrov Monge–Ampère measure of f has
density det A at x: MA_f(closedBall x r) / μ(closedBall x r) → det A as r → 0⁺.
If almost every point x of the interior D of the effective domain has a first-order
expansion of the subdifferential with derivative H x, then there is a measurable S ⊆ D of full
measure in D on which the subdifferential is the single gradient, with the expansion centred at
the gradient and H x positive, and on which H agrees with a measurable map.
The density of the weighted Monge–Ampère measure. Let f : E → EReal be convex (convex
real epigraph, never ⊥) with D the interior of its effective domain and real representative
F, and let H x be, for almost every x ∈ D, a first-order expansion of the subdifferential of
f at x: y' = y + H x (x' - x) + o(‖x' - x‖) for some y, uniformly over the subgradients
y' at points x' near x. By Alexandrov's theorem such H exists, and H x is then the
Alexandrov Hessian of f at x. For every additive Haar measure μ and measurable g, the
Radon–Nikodym derivative of the Monge–Ampère measure MA_{f, g μ}(s) = ∫_{∂f(s ∩ D)} g dμ with
respect to μ is det (H x) * g (∇F x) on D and 0 off D, almost everywhere.
The density of the Monge–Ampère measure is the determinant of the Alexandrov Hessian. Let
f : E → EReal be convex (convex real epigraph, never ⊥), and let H x be, for almost every
x in the interior D of the effective domain, a first-order expansion of the subdifferential of
f at x: y' = y + H x (x' - x) + o(‖x' - x‖) for some y, uniformly over the subgradients
y' at points x' near x. By Alexandrov's theorem such H exists, and H x is then the
Alexandrov Hessian of f at x. For every additive Haar measure μ, the Radon–Nikodym
derivative of the Aleksandrov Monge–Ampère measure MA_f with respect to μ is det (H x) on
D and 0 off D, almost everywhere. This is the case g = 1 of
rnDeriv_mongeAmpereMeasure_withDensity_ae_eq.
Lebesgue decomposition of the weighted Monge–Ampère measure. Let f : E → EReal be convex
(convex real epigraph, never ⊥) with D the interior of its effective domain and real
representative F, and let H x be, for almost every x ∈ D, a first-order expansion of the
subdifferential of f at x (the Alexandrov Hessian, see
rnDeriv_mongeAmpereMeasure_withDensity_ae_eq). Then for every additive Haar measure μ and
measurable g, the Monge–Ampère measure weighted by g μ splits as
MA_{f, g μ} = (MA_{f, g μ})ˢ + det (H x) g (∇F x) dx on D, where (MA_{f, g μ})ˢ is the part
singular with respect to μ.
Lebesgue decomposition of the Monge–Ampère measure. Let f : E → EReal be convex
(convex real epigraph, never ⊥) with D the interior of its effective domain, and let H x be,
for almost every x ∈ D, a first-order expansion of the subdifferential of f at x (the
Alexandrov Hessian, see rnDeriv_mongeAmpereMeasure_ae_eq). Then for every additive Haar measure
μ, the Aleksandrov Monge–Ampère measure splits as MA_f = (MA_f)ˢ + det (H x) dx on D, where
(MA_f)ˢ is the part singular with respect to μ. So the Alexandrov Hessian determines exactly
the absolutely continuous part of MA_f. This is the case g = 1 of
mongeAmpereMeasure_withDensity_eq_singularPart_add_withDensity.
Lebesgue decomposition of the Monge–Ampère measure, with the Alexandrov Hessian. For a
convex f : E → EReal (convex real epigraph, never ⊥) and an additive Haar measure μ, there is
a map H giving, at almost every point x of the effective domain, the positive derivative of the
subdifferential of f, y' = ∇F x + H x (x' - x) + o(‖x' - x‖) with F x = (f x).toReal
(Alexandrov's theorem), and for any such H the Aleksandrov Monge–Ampère measure is
MA_f = (MA_f)ˢ + det (H x) dx on the interior of the effective domain.
An Aleksandrov solution solves the equation almost everywhere. If the Aleksandrov
Monge–Ampère measure of a convex f : E → EReal (convex real epigraph, never ⊥) is absolutely
continuous with an a.e.-measurable density g, that is, f is an Aleksandrov solution of
det D²f = g, then det (H x) = g x for almost every x in the interior D of the effective
domain, where H is the Alexandrov Hessian as in rnDeriv_mongeAmpereMeasure_ae_eq, and g = 0
almost everywhere off D. The converse fails: a cone has Alexandrov Hessian determinant 0 almost
everywhere but singular Monge–Ampère measure (mongeAmpereMeasure_mul_norm_sub).
The Monge–Ampère equation of a transport potential holds almost everywhere. Let
f : E → EReal be convex (convex real epigraph, never ⊥) with D the interior of its effective
domain and real representative F, and let H x be, for almost every x ∈ D, a first-order
expansion of the subdifferential of f at x (the Alexandrov Hessian, see
rnDeriv_mongeAmpereMeasure_withDensity_ae_eq). Let μ be an additive Haar measure, and let
ν₁ ≪ μ have a Lebesgue decomposition with respect to μ (for instance, be σ-finite). If the
Monge–Ampère measure of f weighted by ν₁ is ν₀, that is, f is a weak solution of the
transport equation from ν₀ to ν₁, then the densities g₀ = dν₀/dμ and g₁ = dν₁/dμ
satisfy the Jacobian equation g₀ x = g₁ (∇F x) * det (H x) for almost every x ∈ D, and
g₀ = 0 almost everywhere off D.