The non increasing rearrangement of a measurable function.
The non increasing rearrangement of a measurable function.
Let (Ω, Σ, λ) be a measure space.
The non increasing rearrangement of a measurable function.
Let (Ω, Σ, λ) be a measure space. For each “reasonable”
measurable f : Ω → C,
The non increasing rearrangement of a measurable function.
Let (Ω, Σ, λ) be a measure space. For each “reasonable”
measurable f : Ω → C, the non increasing rearrangement of f is a
non increasing right continuous function f ∗
The non increasing rearrangement of a measurable function.
Let (Ω, Σ, λ) be a measure space. For each “reasonable”
measurable f : Ω → C, the non increasing rearrangement of f is a
non increasing right continuous function f ∗ defined on the interval
(0, ∞), with the property that
The non increasing rearrangement of a measurable function.
Let (Ω, Σ, λ) be a measure space. For each “reasonable”
measurable f : Ω → C, the non increasing rearrangement of f is a
non increasing right continuous function f ∗ defined on the interval
(0, ∞), with the property that f ∗ has the same distribution
function as |f | .
The non increasing rearrangement of a measurable function.
Let (Ω, Σ, λ) be a measure space. For each “reasonable”
measurable f : Ω → C, the non increasing rearrangement of f is a
non increasing right continuous function f ∗ defined on the interval
(0, ∞), with the property that f ∗ has the same distribution
function as |f | .
|{t > 0 : f ∗ (t) > α} = λ ({x ∈ Ω : |f (x)| > α})|
The non increasing rearrangement of a measurable function.
Let (Ω, Σ, λ) be a measure space. For each “reasonable”
measurable f : Ω → C, the non increasing rearrangement of f is a
non increasing right continuous function f ∗ defined on the interval
(0, ∞), with the property that f ∗ has the same distribution
function as |f | .
|{t > 0 : f ∗ (t) > α} = λ ({x ∈ Ω : |f (x)| > α})| for all α > 0 .
The non increasing rearrangement of a measurable function.
Let (Ω, Σ, λ) be a measure space. For each “reasonable”
measurable f : Ω → C, the non increasing rearrangement of f is a
non increasing right continuous function f ∗ defined on the interval
(0, ∞), with the property that f ∗ has the same distribution
function as |f | .
|{t > 0 : f ∗ (t) > α} = λ ({x ∈ Ω : |f (x)| > α})| for all α > 0 .
Consequently,
ˆ ∞
0
ˆ
(f ∗ (t))p dt =
|f (x)|p d µ(x)
Ω
The non increasing rearrangement of a measurable function.
Let (Ω, Σ, λ) be a measure space. For each “reasonable”
measurable f : Ω → C, the non increasing rearrangement of f is a
non increasing right continuous function f ∗ defined on the interval
(0, ∞), with the property that f ∗ has the same distribution
function as |f | .
|{t > 0 : f ∗ (t) > α} = λ ({x ∈ Ω : |f (x)| > α})| for all α > 0 .
Consequently,
ˆ ∞
0
ˆ
(f ∗ (t))p dt =
|f (x)|p d µ(x) for all p > 0 .
Ω
The non increasing rearrangement of a measurable function.
Let (Ω, Σ, λ) be a measure space. For each “reasonable”
measurable f : Ω → C, the non increasing rearrangement of f is a
non increasing right continuous function f ∗ defined on the interval
(0, ∞), with the property that f ∗ has the same distribution
function as |f | .
|{t > 0 : f ∗ (t) > α} = λ ({x ∈ Ω : |f (x)| > α})| for all α > 0 .
Consequently,
ˆ ∞
0
Example:
ˆ
(f ∗ (t))p dt =
|f (x)|p d µ(x) for all p > 0 .
Ω
The non increasing rearrangement of a measurable function.
Let (Ω, Σ, λ) be a measure space. For each “reasonable”
measurable f : Ω → C, the non increasing rearrangement of f is a
non increasing right continuous function f ∗ defined on the interval
(0, ∞), with the property that f ∗ has the same distribution
function as |f | .
|{t > 0 : f ∗ (t) > α} = λ ({x ∈ Ω : |f (x)| > α})| for all α > 0 .
Consequently,
ˆ ∞
0
ˆ
(f ∗ (t))p dt =
|f (x)|p d µ(x) for all p > 0 .
Ω
Example: If f = 2χA + 4χB + 9χC ,
The non increasing rearrangement of a measurable function.
Let (Ω, Σ, λ) be a measure space. For each “reasonable”
measurable f : Ω → C, the non increasing rearrangement of f is a
non increasing right continuous function f ∗ defined on the interval
(0, ∞), with the property that f ∗ has the same distribution
function as |f | .
|{t > 0 : f ∗ (t) > α} = λ ({x ∈ Ω : |f (x)| > α})| for all α > 0 .
Consequently,
ˆ ∞
0
ˆ
(f ∗ (t))p dt =
|f (x)|p d µ(x) for all p > 0 .
Ω
Example: If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Ω of finite measure,
The non increasing rearrangement of a measurable function.
Let (Ω, Σ, λ) be a measure space. For each “reasonable”
measurable f : Ω → C, the non increasing rearrangement of f is a
non increasing right continuous function f ∗ defined on the interval
(0, ∞), with the property that f ∗ has the same distribution
function as |f | .
|{t > 0 : f ∗ (t) > α} = λ ({x ∈ Ω : |f (x)| > α})| for all α > 0 .
Consequently,
ˆ ∞
0
ˆ
(f ∗ (t))p dt =
|f (x)|p d µ(x) for all p > 0 .
Ω
Example: If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Ω of finite measure,
then f ∗ = 9χIC + 4χIB + 2χIA ,
The non increasing rearrangement of a measurable function.
Let (Ω, Σ, λ) be a measure space. For each “reasonable”
measurable f : Ω → C, the non increasing rearrangement of f is a
non increasing right continuous function f ∗ defined on the interval
(0, ∞), with the property that f ∗ has the same distribution
function as |f | .
|{t > 0 : f ∗ (t) > α} = λ ({x ∈ Ω : |f (x)| > α})| for all α > 0 .
Consequently,
ˆ ∞
0
ˆ
(f ∗ (t))p dt =
|f (x)|p d µ(x) for all p > 0 .
Ω
Example: If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Ω of finite measure,
then f ∗ = 9χIC + 4χIB + 2χIA ,
where IC , I B and IA are consecutive intervals, whose lengths equal
λ(C ), λ(B) and λ(A).
The non increasing rearrangement of a measurable function.
Let (Ω, Σ, λ) be a measure space. For each “reasonable”
measurable f : Ω → C, the non increasing rearrangement of f is a
non increasing right continuous function f ∗ defined on the interval
(0, ∞), with the property that f ∗ has the same distribution
function as |f | .
|{t > 0 : f ∗ (t) > α} = λ ({x ∈ Ω : |f (x)| > α})| for all α > 0 .
Consequently,
ˆ ∞
0
ˆ
(f ∗ (t))p dt =
|f (x)|p d µ(x) for all p > 0 .
Ω
Example: If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Ω of finite measure,
then f ∗ = 9χIC + 4χIB + 2χIA ,
where IC , I B and IA are consecutive intervals, whose lengths equal
λ(C ), λ(B) and λ(A).
(Here is a picture:
The non increasing rearrangement of a measurable function.
Let (Ω, Σ, λ) be a measure space. For each “reasonable”
measurable f : Ω → C, the non increasing rearrangement of f is a
non increasing right continuous function f ∗ defined on the interval
(0, ∞), with the property that f ∗ has the same distribution
function as |f | .
|{t > 0 : f ∗ (t) > α} = λ ({x ∈ Ω : |f (x)| > α})| for all α > 0 .
Consequently,
ˆ ∞
0
ˆ
(f ∗ (t))p dt =
|f (x)|p d µ(x) for all p > 0 .
Ω
Example: If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Ω of finite measure,
then f ∗ = 9χIC + 4χIB + 2χIA ,
where IC , I B and IA are consecutive intervals, whose lengths equal
λ(C ), λ(B) and λ(A).
(Here is a picture: We suppose here that λ is two dimensional
Lebesgue measure on some subset Q of R2 .)
If f = 2χA + 4χB + 9χC ,
If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Q,
If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Q,
then f ∗ = 9χIC + 4χIB + 2χIA .
If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Q,
then f ∗ = 9χIC + 4χIB + 2χIA .
where IC , I B and IA are consecutive intervals,
If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Q,
then f ∗ = 9χIC + 4χIB + 2χIA .
where IC , I B and IA are consecutive intervals,
whose lengths equal λ(C ), λ(B) and λ(A).
If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Q,
then f ∗ = 9χIC + 4χIB + 2χIA .
where IC , I B and IA are consecutive intervals,
whose lengths equal λ(C ), λ(B) and λ(A).
If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Q,
then f ∗ = 9χIC + 4χIB + 2χIA .
where IC , I B and IA are consecutive intervals,
whose lengths equal λ(C ), λ(B) and λ(A).
If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Q,
then f ∗ = 9χIC + 4χIB + 2χIA .
where IC , I B and IA are consecutive intervals,
whose lengths equal λ(C ), λ(B) and λ(A).
If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Q,
then f ∗ = 9χIC + 4χIB + 2χIA .
where IC , I B and IA are consecutive intervals,
whose lengths equal λ(C ), λ(B) and λ(A).
If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Q,
then f ∗ = 9χIC + 4χIB + 2χIA .
where IC , I B and IA are consecutive intervals,
whose lengths equal λ(C ), λ(B) and λ(A).
If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Q,
then f ∗ = 9χIC + 4χIB + 2χIA .
where IC , I B and IA are consecutive intervals,
whose lengths equal λ(C ), λ(B) and λ(A).
If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Q,
then f ∗ = 9χIC + 4χIB + 2χIA .
where IC , I B and IA are consecutive intervals,
whose lengths equal λ(C ), λ(B) and λ(A).
If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Q,
then f ∗ = 9χIC + 4χIB + 2χIA .
where IC , I B and IA are consecutive intervals,
whose lengths equal λ(C ), λ(B) and λ(A).
If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Q,
then f ∗ = 9χIC + 4χIB + 2χIA .
where IC , I B and IA are consecutive intervals,
whose lengths equal λ(C ), λ(B) and λ(A).
If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Q,
then f ∗ = 9χIC + 4χIB + 2χIA .
where IC , I B and IA are consecutive intervals,
whose lengths equal λ(C ), λ(B) and λ(A).
If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Q,
then f ∗ = 9χIC + 4χIB + 2χIA .
where IC , I B and IA are consecutive intervals,
whose lengths equal λ(C ), λ(B) and λ(A).
If f = 2χA + 4χB + 9χC ,
where A, B and C are disjoint subsets of Q,
then f ∗ = 9χIC + 4χIB + 2χIA .
where IC , I B and IA are consecutive intervals,
whose lengths equal λ(C ), λ(B) and λ(A).