Introduction Statement and Sketch Disintegration of LN Examples
A proof of Sudakov theorem, with strictly convex
norms, via disintegration of measures
Preprint
Laura Caravenna
SISSA-ISAS
First Winter School at imdea, Madrid
January 26-30, 2009
Sudakov Theorem with Strictly Convex Norms
1 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sudakov Theorem with Strictly Convex Norms
1. Introduction
Disintegration of Probability Measures
Monge-Kantorovich Problem in RN
2. Statement and Sketch
3. Disintegration of the Lebesgue Measure
4. Examples
Sudakov Theorem with Strictly Convex Norms
2 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sudakov Theorem with Strictly Convex Norms
1. Introduction
Disintegration of Probability Measures
Monge-Kantorovich Problem in RN
2. Statement and Sketch
3. Disintegration of the Lebesgue Measure
4. Examples
Sudakov Theorem with Strictly Convex Norms
3 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Disintegration of Probability Measures
(Z , Σ, λ) probability space, p : Z 7→ V . Then we define
Sudakov Theorem with Strictly Convex Norms
4 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Disintegration of Probability Measures
(Z , Σ, λ) probability space, p : Z 7→ V . Then we define
◮
Θ = p♯ Σ :
Q ∈Θ
Sudakov Theorem with Strictly Convex Norms
⇔
p −1 (Q) ∈ Σ
4 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Disintegration of Probability Measures
(Z , Σ, λ) probability space, p : Z 7→ V . Then we define
◮
Θ = p♯ Σ :
◮
θ = p♯ λ :
Q ∈Θ
⇔
p −1 (Q) ∈ Σ
θ(Q) := λ(p −1 (Q)) ∀Q ∈ Θ
Sudakov Theorem with Strictly Convex Norms
4 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Disintegration of Probability Measures
(Z , Σ, λ) probability space, p : Z 7→ V . Then we define
◮
Θ = p♯ Σ :
◮
θ = p♯ λ :
Q ∈Θ
⇔
p −1 (Q) ∈ Σ
θ(Q) := λ(p −1 (Q)) ∀Q ∈ Θ
Definition: a disintegration of λ w.r.t. p
A family of probability measures λv v ∈V on (Z , Σ), s.t.
◮
λ· (B) θ-measurable, ∀B ∈ Σ,
◮
for all B ∈ Σ, S ∈ Θ
λ(B ∩ p
−1
(S)) =
Z
λv (B)dθ(v )
S
Sudakov Theorem with Strictly Convex Norms
4 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Disintegration of Probability Measures
(Z , Σ, λ) probability space, p : Z 7→ V . Then we define
◮
Θ = p♯ Σ :
◮
θ = p♯ λ :
Q ∈Θ
⇔
p −1 (Q) ∈ Σ
θ(Q) := λ(p −1 (Q)) ∀Q ∈ Θ
Definition: a disintegration of λ w.r.t. p
A family of probability measures λv v ∈V on (Z , Σ), s.t.
◮
λ· (B) θ-measurable, ∀B ∈ Σ,
◮
for all B ∈ Σ, S ∈ Θ
λ(B ∩ p
−1
(S)) =
Z
λv (B)dθ(v )
S
A disintegration is subordinated to p if
λv carried by p −1 (v ),
for θ-a.e. v
Sudakov Theorem with Strictly Convex Norms
4 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Disintegration Theorems
p : (Z , Σ, λ) 7→ (V , Θ, θ)
R
p −1 (S)
q dλ =
Sudakov Theorem with Strictly Convex Norms
R R
S
Z
q dλv dθ(v )
5 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Disintegration Theorems
p : (Z , Σ, λ) 7→ (V , Θ, θ)
Example
R
p −1 (S)
q dλ =
R R
S
Z
q dλv dθ(v )
Z product space, p projection onto the first factor space
Y
π ∈ P(X × Y )
p : (x, y ) 7→ x
S
π(S) =
Z
πx (S) dµ(y )
µ = p ♯π
X
Sudakov Theorem with Strictly Convex Norms
5 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Disintegration Theorems
p : (Z , Σ, λ) 7→ (V , Θ, θ)
Example
R
p −1 (S)
q dλ =
R R
S
Z
q dλv dθ(v )
Z product space, p projection onto the first factor space
Y
π ∈ P(X × Y )
p : (x, y ) 7→ x
S
π(S) =
Z
πx (S) dµ(y )
µ = p ♯π
X
Sudakov Theorem with Strictly Convex Norms
5 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Disintegration Theorems
p : (Z , Σ, λ) 7→ (V , Θ, θ)
Example
R
p −1 (S)
q dλ =
R R
S
Z
q dλv dθ(v )
Z product space, p projection onto the first factor space
Y
π ∈ P(X × Y )
p : (x, y ) 7→ x
S
π(S) =
Z
πx (S) dµ(y )
µ = p ♯π
X
Sudakov Theorem with Strictly Convex Norms
5 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Disintegration Theorems
p : (Z , Σ, λ) 7→ (V , Θ, θ)
Example
R
p −1 (S)
q dλ =
R R
S
Z
q dλv dθ(v )
Z product space, p projection onto the first factor space
Y
π ∈ P(X × Y )
p : (x, y ) 7→ x
S
π(S) =
Z
πx (S) dµ(y )
µ = p ♯π
X
Sudakov Theorem with Strictly Convex Norms
5 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Disintegration Theorems
p : (Z , Σ, λ) 7→ (V , Θ, θ)
R
p −1 (S)
q dλ =
R R
S
q
dλ
dθ(v )
v
Z
Theorem
Σ σ-generated by a countable family =⇒ existence and uniqueness
Theorem
Θ has a sub-σ-algebra generated by a countable family and
containing points
=⇒
subordinated
Sudakov Theorem with Strictly Convex Norms
5 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Monge Problem (1781)
Transport map from (X , µ) to (Y , ν):
a function t : X 7→ Y such that
X
Y
Sudakov Theorem with Strictly Convex Norms
6 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Monge Problem (1781)
Transport map from (X , µ) to (Y , ν):
a function t : X 7→ Y such that
ν is the image measure of µ
Y
t
E
t −1 (E )
ν(E ) = µ(t(E )) ∀E ∈ B(Y )
X
Sudakov Theorem with Strictly Convex Norms
6 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Monge Problem (1781)
Transport map from (X , µ) to (Y , ν):
a function t : X 7→ Y such that
ν is the image measure of µ
Cost of a transport map:
Z
W (t) := kt(x) − xkdµ(x)
X
Sudakov Theorem with Strictly Convex Norms
6 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Monge Problem (1781)
Transport map from (X , µ) to (Y , ν):
a function t : X 7→ Y such that
ν is the image measure of µ
Cost of a transport map:
Z
W (t) := kt(x) − xkdµ(x)
X
Problem
To minimize W (t),
Sudakov Theorem with Strictly Convex Norms
t transport map
6 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Kantorovich Formulation (1942)
Transport plan from (X , µ) to (Y , ν):
a measure π ∈ P(X × Y )
Y
ν
µ
X
Sudakov Theorem with Strictly Convex Norms
7 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Kantorovich Formulation (1942)
Transport plan from (X , µ) to (Y , ν):
a measure π ∈ P(X × Y )
having marginals µ, ν
Y
ν(E ) = π(X × E )
∀E ∈ B(Y )
ν
µ
X
Sudakov Theorem with Strictly Convex Norms
7 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Kantorovich Formulation (1942)
Transport plan from (X , µ) to (Y , ν):
a measure π ∈ P(X × Y )
having marginals µ, ν
Y
Cost of a transport plan:
Z
ky − xkdπ(x, y )
W (π) :=
ν
X ×Y
µ
X
Sudakov Theorem with Strictly Convex Norms
7 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Kantorovich Formulation (1942)
Transport plan from (X , µ) to (Y , ν):
a measure π ∈ P(X × Y )
having marginals µ, ν
Y
Cost of a transport plan:
Z
ky − xkdπ(x, y )
W (π) :=
ν
X ×Y
µ
X
Problem
To minimize W (π),
Sudakov Theorem with Strictly Convex Norms
π transport plan
7 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Comparison
We have a minimization problem, the variable is
Sudakov Theorem with Strictly Convex Norms
8 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Comparison
We have a minimization problem, the variable is
Monge: t ∈ T (µ, ν)
Kantorovich: π ∈ Π(µ, ν)
Y
Y
ν
ν
µ
t 7→
Z
µ
X
kt(x) − xk dµ(x)
π 7→
Sudakov Theorem with Strictly Convex Norms
Z
X
ky − xk dπ(x, y )
8 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Comparison
We have a minimization problem, the variable is
Kantorovich: π ∈ Π(µ, ν)
Monge: t ∈ T (µ, ν)
Y
Y
B
t(x)
x
x 7→ t(x)
x
X
X
P({x 7→ y ∈ B}) = πx (B)
Sudakov Theorem with Strictly Convex Norms
8 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sudakov Theorem with Strictly Convex Norms
1. Introduction
Disintegration of Probability Measures
Monge-Kantorovich Problem in RN
2. Statement and Sketch
3. Disintegration of the Lebesgue Measure
4. Examples
Sudakov Theorem with Strictly Convex Norms
9 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Statement
Sudakov Theorem (stated in 1976)
Consider a strictly convex norm || · || on RN , possibly asymmetric.
Be given
µ = f dLN ,
ν
probability measures on RN . Then there is a transport map t s.t.
Z
Z
||y − x|| dπ.
||t(x) − x|| dµ = min
RN
π∈Π(µ,ν)
RN ×RN
Sudakov Theorem with Strictly Convex Norms
10 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Statement
Sudakov Theorem (stated in 1976)
Consider a strictly convex norm || · || on RN , possibly asymmetric.
Be given
ν
µ = f dLN ,
probability measures on RN . Then there is a transport map t s.t.
Z
Z
||y − x|| dπ.
||t(x) − x|| dµ = min
RN
π∈Π(µ,ν)
RN ×RN
Sudakov Theorem with Strictly Convex Norms
10 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Statement
Sudakov Theorem (stated in 1976)
Consider a strictly convex norm || · || on RN , possibly asymmetric.
Be given
µ = f dLN ,
ν
probability measures on RN . Then there is a transport map t s.t.
Z
Z
||y − x|| dπ.
||t(x) − x|| dµ = min
RN
π∈Π(µ,ν)
RN ×RN
Remarks
Consider the vector field giving the direction of the transport.
Sudakov Theorem with Strictly Convex Norms
10 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Statement
Sudakov Theorem (stated in 1976)
Consider a strictly convex norm || · || on RN , possibly asymmetric.
Be given
µ = f dLN ,
ν
probability measures on RN . Then there is a transport map t s.t.
Z
Z
||y − x|| dπ.
||t(x) − x|| dµ = min
RN
π∈Π(µ,ν)
RN ×RN
Remarks
Consider the vector field giving the direction of the transport.
◮
its divergence is a series of measures
Sudakov Theorem with Strictly Convex Norms
10 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Statement
Sudakov Theorem (stated in 1976)
Consider a strictly convex norm || · || on RN , possibly asymmetric.
Be given
µ = f dLN ,
ν
probability measures on RN . Then there is a transport map t s.t.
Z
Z
||y − x|| dπ.
||t(x) − x|| dµ = min
RN
π∈Π(µ,ν)
RN ×RN
Remarks
Consider the vector field giving the direction of the transport.
◮
its divergence is a series of measures
Sudakov Theorem with Strictly Convex Norms
10 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Statement
Sudakov Theorem (stated in 1976)
Consider a strictly convex norm || · || on RN , possibly asymmetric.
Be given
ν = g dLN
µ = f dLN ,
probability measures on RN . Then there is a transport map t s.t.
Z
Z
||y − x|| dπ.
||t(x) − x|| dµ = min
RN
π∈Π(µ,ν)
RN ×RN
Remarks
Consider the vector field giving the direction of the transport.
◮
its divergence is a series of measures
Sudakov Theorem with Strictly Convex Norms
10 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
N=1
N=1
Sudakov Theorem with Strictly Convex Norms
The statement
holds. Consider
G (x) = µ((−∞, x)),
11 / 18
F (x) = ν((−∞, x)).
Then, the an optimal map is given by
t(x) = sup y ∈ R : F (y ) ≤ G (x) .
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
N=1
The statement holds. Consider
G (x) = µ((−∞, x)),
F (x) = ν((−∞, x)).
Then, the an optimal map is given by
t(x) = sup y ∈ R : F (y ) ≤ G (x) .
Uniqueness
Sudakov Theorem with Strictly Convex Norms
11 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
N=1
The statement holds. Consider
G (x) = µ((−∞, x)),
F (x) = ν((−∞, x)).
Then, the an optimal map is given by
t(x) = sup y ∈ R : F (y ) ≤ G (x) .
Uniqueness
Recovered with additional conditions
◮
monotonicity on rays
Sudakov Theorem with Strictly Convex Norms
11 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Statement
Consider
◮
(RN , k·k), with k·k strictly convex
◮
µ = f dLN , ν = g dLN
Then there exists an optimal map.
Sudakov Theorem with Strictly Convex Norms
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Statement
Consider
(RN , k·k)
k·k strictly conv.
µ = f dLN
ν = g dL
Proof.
1. The mass moves along lines
N
=⇒
∃ optimal map
N = 1 OK
Sudakov Theorem with Strictly Convex Norms
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Statement
Consider
(RN , k·k)
Proof.
1. The mass moves along lines
k·k strictly conv.
µ = f dLN
ν = g dLN
=⇒
∃ optimal map
N = 1 OK
Sudakov Theorem with Strictly Convex Norms
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Statement
Consider
(RN , k·k)
Proof.
1. The mass moves along lines
k·k strictly conv.
µ = f dLN
ν = g dLN
=⇒
∃ optimal map
N = 1 OK
Sudakov Theorem with Strictly Convex Norms
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Statement
Consider
(RN , k·k)
Proof.
1. The mass moves along lines
k·k strictly conv.
µ = f dLN
ν = g dLN
=⇒
∃ optimal map
N = 1 OK
Sudakov Theorem with Strictly Convex Norms
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Statement
Consider
(RN , k·k)
Proof.
1. The mass moves along lines
k·k strictly conv.
µ = f dLN
ν = g dLN
=⇒
∃ optimal map
N = 1 OK
Sudakov Theorem with Strictly Convex Norms
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Statement
Consider
(RN , k·k)
Proof.
1. The mass moves along lines
k·k strictly conv.
µ = f dLN
ν = g dLN
=⇒
∃ optimal map
N = 1 OK
Sudakov Theorem with Strictly Convex Norms
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Statement
Consider
Proof.
1. The mass moves along rays
N
(R , k·k)
k·k strictly conv.
µ = f dLN
ν = g dLN
R(x)
=⇒
∃ optimal map
N = 1 OK
Sudakov Theorem with Strictly Convex Norms
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Statement
Consider
Proof.
1. The mass moves along rays
N
(R , k·k)
k·k strictly conv.
µ = f dLN
ν = g dLN
R(x)
=⇒
∃ optimal map
N = 1 OK
Sudakov Theorem with Strictly Convex Norms
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Statement
Consider
Proof.
1. The mass moves along rays
N
(R , k·k)
k·k strictly conv.
µ = f dLN
ν = g dLN
=⇒
∃ optimal map
R(x)
Consider the set where mass can be moved
T = ∪x:
R(x)6={x} R(x)
N = 1 OK
Sudakov Theorem with Strictly Convex Norms
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Proof.
Statement
1. Rays are invariant sets for the transport
Consider
2. Rays ‘partition’ the transport set
(RN , k·k)
k·k strictly conv.
µ = f dLN
ν = g dLN
=⇒
∃ optimal map
N = 1 OK
Sudakov Theorem with Strictly Convex Norms
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Proof.
Statement
1. Rays are invariant sets for the transport
Consider
2. Rays ‘partition’ the transport set
(RN , k·k)
3. Disintegrate µ, ν w.r.t. the rays
k·k strictly conv.
µ = f dLN
ν = g dLN
=⇒
∃ optimal map
N = 1 OK
Sudakov Theorem with Strictly Convex Norms
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Proof.
Statement
1. Rays are invariant sets for the transport
Consider
2. Rays ‘partition’ the transport set
(RN , k·k)
3. Disintegrate µ, ν w.r.t. the rays
µα να on each ray
k·k strictly conv.
µ = f dLN
ν = g dLN
=⇒
∃ optimal map
N = 1 OK
Sudakov Theorem with Strictly Convex Norms
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Proof.
Statement
1. Rays are invariant sets for the transport
Consider
2. Rays ‘partition’ the transport set
(RN , k·k)
3. Disintegrate µ, ν w.r.t. the rays
µα να on each ray
k·k strictly conv.
µ = f dLN
4. Solve the 1-d problem on each ray
ν = g dLN
=⇒
∃ optimal map
N = 1 OK
Sudakov Theorem with Strictly Convex Norms
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Proof.
Statement
1. Rays are invariant sets for the transport
Consider
2. Rays ‘partition’ the transport set
(RN , k·k)
3. Disintegrate µ, ν w.r.t. the rays
µα να on each ray
k·k strictly conv.
µ = f dLN
4. Solve the 1-d problem on each ray
tα
ν = g dLN
=⇒
∃ optimal map
N = 1 OK
Sudakov Theorem with Strictly Convex Norms
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Proof.
Statement
1. Rays are invariant sets for the transport
Consider
2. Rays ‘partition’ the transport set
(RN , k·k)
3. Disintegrate µ, ν w.r.t. the rays
µα να on each ray
k·k strictly conv.
µ = f dLN
ν = g dLN
4. Solve the 1-d problem on each ray
tα
5. Glue together the maps
=⇒
∃ optimal map
N = 1 OK
Sudakov Theorem with Strictly Convex Norms
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Proof.
Statement
1. Rays are invariant sets for the transport
Consider
2. Rays ‘partition’ the transport set
(RN , k·k)
3. Disintegrate µ, ν w.r.t. the rays
µα να on each ray
k·k strictly conv.
µ = f dLN
ν = g dLN
=⇒
∃ optimal map
N = 1 OK
4. Solve the 1-d problem on each ray
5. Glue together the maps
6. Compare with other transport plans
π ∈ Π(µ, ν)
πα ∈ Π(µα , να )
tα
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Proof.
Statement
1. Rays are invariant sets for the transport
Consider
2. Rays ‘partition’ the transport set
(RN , k·k)
3. Disintegrate µ, ν w.r.t. the rays
µα να on each ray
k·k strictly conv.
µ = f dLN
ν = g dLN
=⇒
4. Solve the 1-d problem on each ray
5. Glue together the maps
6. Compare with other transport plans
π ∈ Π(µ, ν)
∃ optimal map
N = 1 OK
tα
Z
R2α
ky − xkdπα ≥
Sudakov Theorem with Strictly Convex Norms
πα ∈ Π(µα , να )
Z
ktα (x) − xkdµα
Rα
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch of Sudakov Theorem
Proof.
Statement
1. Rays are invariant sets for the transport
Consider
2. Rays ‘partition’ the transport set
(RN , k·k)
3. Disintegrate µ, ν w.r.t. the rays
µα να on each ray
k·k strictly conv.
µ = f dLN
ν = g dLN
=⇒
4. Solve the 1-d problem on each ray
5. Glue together the maps
6. Compare with other transport plans
π ∈ Π(µ, ν)
∃ optimal map
N = 1 OK
tα
Z
R2α
ky − xkdπα ≥
Sudakov Theorem with Strictly Convex Norms
πα ∈ Π(µα , να )
Z
ktα (x) − xkdµα
Rα
12 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sudakov Theorem with Strictly Convex Norms
1. Introduction
Disintegration of Probability Measures
Monge-Kantorovich Problem in RN
2. Statement and Sketch
3. Disintegration of the Lebesgue Measure
4. Examples
Sudakov Theorem with Strictly Convex Norms
13 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch
Forget the endpoints. Partition of T into model sets
a(x)
x
h−
b(x)
t
Sudakov Theorem with Strictly Convex Norms
h+
14 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch
Forget the endpoints. Partition of T into model sets
s
a(x)
b(x)
Sudakov Theorem with Strictly Convex Norms
14 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch
Forget the endpoints. Partition of T into model sets
a(x)
h−
Z
Z
N
φ dL =
x
b(x)
t
h+
h−
dx1
Z
T ∩Hx1
h+
φ dx2 . . . dxN
Introduction Statement and Sketch Disintegration of LN Examples
Sketch
Forget the endpoints. Partition of T into model sets
a(x)
x
h−
Z
Z
N
φ dL =
b(x)
h+
t
h+
h−
dx1
Z
= ...
Z
Z
N−1
dH
(y )
=
Z
φ dx2 . . . dxN
T ∩Hx1
h+
h−
αx1 (y )φ dx1
Sudakov Theorem with Strictly Convex Norms
14 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sketch
Forget the endpoints. Partition of T into model sets
a(x)
x
h−
Z
Z
N
φ dL =
b(x)
h+
t
h+
h−
dx1
Z
= ...
Z
Z
N−1
dH
(y )
=
Z
φ dx2 . . . dxN
T ∩Hx1
h+
h−
αx1 (y )φ dx1
The disintegrated measures are absolutely continuous w.r.t. L1 !
Sudakov Theorem with Strictly Convex Norms
14 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Fundamental estimate: absolutely continuous push forward
h−
s
t
Sudakov Theorem with Strictly Convex Norms
h+
15 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Fundamental estimate: absolutely continuous push forward
h−
s
t
h+
For all T ′ ⊂ T made of rays
Sudakov Theorem with Strictly Convex Norms
15 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Fundamental estimate: absolutely continuous push forward
h−
s
For all T ′ ⊂ T made of rays

N−1 
 h++ −t
Ts′ ≤ Tt′ h −s
t
h+
for h− ≤ s ≤ t < h+


Sudakov Theorem with Strictly Convex Norms
15 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Fundamental estimate: absolutely continuous push forward
h−
s
For all T ′ ⊂ T made of rays

N−1 
 h++ −t
Ts′ ≤ Tt′ h −s
t
h+
for h− ≤ s ≤ t < h+


Sudakov Theorem with Strictly Convex Norms
15 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Fundamental estimate: absolutely continuous push forward
h−
s
For all T ′ ⊂ T made of rays

N−1 
 h++ −t
Ts′ ≤ Tt′ h −s
T ′ ≤ t−h− N−1 T ′ 
−
s
t
s−h
t
h+
for h− ≤ s ≤ t < h+
for h− < s ≤ t ≤ h+
Sudakov Theorem with Strictly Convex Norms
15 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Fundamental estimate: absolutely continuous push forward
h−
s
For all T ′ ⊂ T made of rays

N−1 
 h++ −t
Ts′ ≤ Tt′ h −s
T ′ ≤ t−h− N−1 T ′ 
−
s
t
s−h
t
h+
for h− ≤ s ≤ t < h+
for h− < s ≤ t ≤ h+
Corollary
The set of endpoints is LN -negligible
Sudakov Theorem with Strictly Convex Norms
15 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Sudakov Theorem with Strictly Convex Norms
1. Introduction
Disintegration of Probability Measures
Monge-Kantorovich Problem in RN
2. Statement and Sketch
3. Disintegration of the Lebesgue Measure
4. Examples
Sudakov Theorem with Strictly Convex Norms
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Introduction Statement and Sketch Disintegration of LN Examples
Theorem
There exist a Borel set MN ⊂ [−1, 1]3 with |[−1, 1]3 \ MN | = 0
and a Borel map f : MN → [−2, 2]2 × [−2, 2]2 such that the
following holds. If we define for x ∈ MN the open segment lx
connecting (f1 (x), −2) to (f2 (x), 2), then
◮
{x} = lx ∩ MN for all x ∈ MN ,
◮ lx
∩ ly = ∅ for all x, y ∈ MN different.
Sudakov Theorem with Strictly Convex Norms
17 / 18
Introduction Statement and Sketch Disintegration of LN Examples
Theorem
There exist a Borel set MN ⊂ [−1, 1]3 with |[−1, 1]3 \ MN | = 0
and a Borel map f : MN → [−2, 2]2 × [−2, 2]2 such that the
following holds. If we define for x ∈ MN the open segment lx
connecting (f1 (x), −2) to (f2 (x), 2), then
◮
{x} = lx ∩ MN for all x ∈ MN ,
◮ lx
∩ ly = ∅ for all x, y ∈ MN different.
Example (Transport rays do not fill the space)
Sudakov Theorem with Strictly Convex Norms
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Monge ill posed
Example
µ = δx0
ν=
1
δy1 + δy2
2
We can have that
◮
there is no transport map
Sudakov Theorem with Strictly Convex Norms
18 / 18
Monge ill posed
Example
µ = H1 ↾[0,1]⊗{0}
ν = 0, 5 · H1 ↾[0,1]⊗{1}
+ H1 ↾[0,1]⊗{−1}
We can have that
◮
there is no transport map
◮
the minimum is not attained
Sudakov Theorem with Strictly Convex Norms
18 / 18
Monge ill posed
Example
µ = H1 ↾[0,1]⊗{0}
ν = 0, 5 · H1 ↾[0,1]⊗{1}
+ H1 ↾[0,1]⊗{−1}
We can have that
◮
there is no transport map
◮
the minimum is not attained
Sudakov Theorem with Strictly Convex Norms
18 / 18
Monge ill posed
Example
µ = H1 ↾[0,1]⊗{0}
ν = 0, 5 · H1 ↾[0,1]⊗{1}
+ H1 ↾[0,1]⊗{−1}
We can have that
◮
there is no transport map
◮
the minimum is not attained
Sudakov Theorem with Strictly Convex Norms
18 / 18
Monge ill posed
Example
µ = H1 ↾[0,1]⊗{0}
ν = 0, 5 · H1 ↾[0,1]⊗{1}
+ H1 ↾[0,1]⊗{−1}
We can have that
◮
there is no transport map
◮
the minimum is not attained
Sudakov Theorem with Strictly Convex Norms
18 / 18
Monge ill posed
We can have that
◮
there is no transport map
◮
the minimum is not attained
Sudakov Theorem with Strictly Convex Norms
18 / 18
Monge ill posed
We can have that
◮
there is no transport map
◮
the minimum is not attained
Theorem
When c is l.s.c., then Monge-Kantorovich problem has a minimizer.
Sudakov Theorem with Strictly Convex Norms
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