on the approximate variational measure

on the approximate variational measure

Strona główna
Strona tytułowa
ON THE APPROXIMATE VARIATIONAL
MEASURE
Piotr Sworowski
Casimirus the Great University
Bydgoszcz, Poland
Spis treści
JJ
II
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Strona 1 z 41
Powrót
Sugar Cane Symposium in Real Analysis XXXVII
São Carlos, 3–6.06.2013
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SOME DEFINITIONS
Strona tytułowa
Spis treści
By a tagged interval we mean a pair (I, x), where x ∈ I ⊂ R.
JJ
II
J
I
By a division in an interval [a, b] we mean a finite collection {(Ii, xi)}ki=1 of tagged
intervals, where intervals Ii therein are pairwise nonoverlapping.
A division in [a, b] is called a partition of [a, b] if
k
[
Ii = [a, b].
Strona 2 z 41
i=1
Powrót
Having a function δ : R → (0, ∞), called a gauge, we say that a division
{(Ii, xi)}ki=1 is δ-fine if for each i,
Pełny ekran
Ii ⊂ (xi − δ(xi), xi + δ(xi)).
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We say a division {(Ii, xi)}ki=1 is anchored in a set E ⊂ R if xi ∈ E for each i.
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Strona główna
SOME DEFINITIONS
Strona tytułowa
Spis treści
By a tagged interval we mean a pair (I, x), where x ∈ I ⊂ R.
JJ
II
J
I
By a division in an interval [a, b] we mean a finite collection {(Ii, xi)}ki=1 of tagged
intervals, where intervals Ii therein are pairwise nonoverlapping.
A division in [a, b] is called a partition of [a, b] if
k
[
Ii = [a, b].
Strona 3 z 41
i=1
Powrót
Having a function δ : R → (0, ∞), called a gauge, we say that a division
{(Ii, xi)}ki=1 is δ-fine if for each i,
Pełny ekran
Ii ⊂ (xi − δ(xi), xi + δ(xi)).
Zamknij
We say a division {(Ii, xi)}ki=1 is anchored in a set E ⊂ R if xi ∈ E for each i.
Koniec
Strona główna
SOME DEFINITIONS
Strona tytułowa
Spis treści
By a tagged interval we mean a pair (I, x), where x ∈ I ⊂ R.
JJ
II
J
I
By a division in an interval [a, b] we mean a finite collection {(Ii, xi)}ki=1 of tagged
intervals, where intervals Ii therein are pairwise nonoverlapping.
A division in [a, b] is called a partition of [a, b] if
k
[
Ii = [a, b].
Strona 4 z 41
i=1
Powrót
Having a function δ : R → (0, ∞), called a gauge, we say that a division
{(Ii, xi)}ki=1 is δ-fine if for each i,
Pełny ekran
Ii ⊂ (xi − δ(xi), xi + δ(xi)).
Zamknij
We say a division {(Ii, xi)}ki=1 is anchored in a set E ⊂ R if xi ∈ E for each i.
Koniec
Strona główna
SOME DEFINITIONS
Strona tytułowa
Spis treści
By a tagged interval we mean a pair (I, x), where x ∈ I ⊂ R.
JJ
II
J
I
By a division in an interval [a, b] we mean a finite collection {(Ii, xi)}ki=1 of tagged
intervals, where intervals Ii therein are pairwise nonoverlapping.
A division in [a, b] is called a partition of [a, b] if
k
[
Ii = [a, b].
Strona 5 z 41
i=1
Powrót
Having a function δ : R → (0, ∞), called a gauge, we say that a division
{(Ii, xi)}ki=1 is δ-fine if for each i,
Pełny ekran
Ii ⊂ (xi − δ(xi), xi + δ(xi)).
Zamknij
We say a division {(Ii, xi)}ki=1 is anchored in a set E ⊂ R if xi ∈ E for each i.
Koniec
Strona główna
SOME DEFINITIONS
Strona tytułowa
Spis treści
By a tagged interval we mean a pair (I, x), where x ∈ I ⊂ R.
JJ
II
J
I
By a division in an interval [a, b] we mean a finite collection {(Ii, xi)}ki=1 of tagged
intervals, where intervals Ii therein are pairwise nonoverlapping.
A division in [a, b] is called a partition of [a, b] if
k
[
Ii = [a, b].
Strona 6 z 41
i=1
Powrót
Having a function δ : R → (0, ∞), called a gauge, we say that a division
{(Ii, xi)}ki=1 is δ-fine if for each i,
Pełny ekran
Ii ⊂ (xi − δ(xi), xi + δ(xi)).
Zamknij
We say a division {(Ii, xi)}ki=1 is anchored in a set E ⊂ R if xi ∈ E for each i.
Koniec
Strona główna
KURZWEIL–HENSTOCK INTEGRAL
Strona tytułowa
Spis treści
DEFINITION.
Kurzweil 1957 & Henstock
R b 1960
We call a function f : [a, b] → R, H-integrable, with the integral I = (H) a f ∈ R,
if for each ε > 0 there exists a gauge δ, such that for every δ-fine partition
{(Ii, xi)}ki=1 of [a, b],
k
X
f (xi)|Ii| − I < ε.
i=1
JJ
II
J
I
Strona 7 z 41
Set the indefinite integral of f :
Powrót
Z
F (x) = (H)
x
f,
a
x ∈ [a, b].
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How to characterize F ?
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KURZWEIL–HENSTOCK INTEGRAL
Strona tytułowa
Spis treści
DEFINITION.
Kurzweil 1957 & Henstock
R b 1960
We call a function f : [a, b] → R, H-integrable, with the integral I = (H) a f ∈ R,
if for each ε > 0 there exists a gauge δ, such that for every δ-fine partition
{(Ii, xi)}ki=1 of [a, b],
k
X
f (xi)|Ii| − I < ε.
i=1
JJ
II
J
I
Strona 8 z 41
Set the indefinite integral of f :
Powrót
Z
F (x) = (H)
x
f,
a
x ∈ [a, b].
Pełny ekran
How to characterize F ?
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Strona główna
VARIATIONAL MEASURE
Strona tytułowa
Spis treści
Let F : [a, b] → R.
By |E|F we mean the variational measure of E ⊂ [a, b] induced by F ; i.e.,
|E|F = inf sup
δ
P
k
X
|∆F (Ii)|,
JJ
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J
I
i=1
where sup runs over all δ-fine divisions {(Ii, xi)}ki=1 anchored in E.
Strona 9 z 41
The function F is said to be SL (after Strong Lusin Condition) if | · |F is absolutely
continuous; i.e.,
|E|F = 0 for every (Lebesgue) nullset E ⊂ [a, b].
Powrót
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VARIATIONAL MEASURE
Strona tytułowa
Spis treści
Let F : [a, b] → R.
By |E|F we mean the variational measure of E ⊂ [a, b] induced by F ; i.e.,
|E|F = inf sup
δ
P
k
X
|∆F (Ii)|,
JJ
II
J
I
i=1
where sup runs over all δ-fine divisions {(Ii, xi)}ki=1 anchored in E.
Strona 10 z 41
The function F is said to be SL (after Strong Lusin Condition) if | · |F is absolutely
continuous; i.e.,
|E|F = 0 for every (Lebesgue) nullset E ⊂ [a, b].
Powrót
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A CHARACTERIZATION OF H-INTEGRAL
Spis treści
THEOREM. Ene 1994, Bongiorno & Di Piazza & Skvortsov 1995.
Let F : [a, b] → R, F (a) = 0. TAE:
❶ F is an indefinite Kurzweil–Henstock integral (of F 0 ),
JJ
II
J
I
❷ F is SL.
Strona 11 z 41
Key steps of the proof:
Powrót
• notice that | · |F is absolutely continuous on each set
{x ∈ [a, b] : F 0(x) exists and F 0(x) ≤ n},
n ∈ N;
Pełny ekran
so it’s enough to prove
• | · |F is absolutely continuous =⇒ F is almost everywhere differentiable.
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Strona główna
Strona tytułowa
A CHARACTERIZATION OF H-INTEGRAL
Spis treści
THEOREM. Ene 1994, Bongiorno & Di Piazza & Skvortsov 1995.
Let F : [a, b] → R, F (a) = 0. TAE:
❶ F is an indefinite Kurzweil–Henstock integral (of F 0 ),
JJ
II
J
I
❷ F is SL.
Strona 12 z 41
Key steps of the proof:
Powrót
• notice that | · |F is absolutely continuous on each set
{x ∈ [a, b] : F 0(x) exists and F 0(x) ≤ n},
n ∈ N;
Pełny ekran
so it’s enough to prove
• | · |F is absolutely continuous =⇒ F is almost everywhere differentiable.
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Strona główna
Strona tytułowa
A CHARACTERIZATION OF H-INTEGRAL
Spis treści
THEOREM. Ene 1994, Bongiorno & Di Piazza & Skvortsov 1995.
Let F : [a, b] → R, F (a) = 0. TAE:
❶ F is an indefinite Kurzweil–Henstock integral (of F 0 ),
JJ
II
J
I
❷ F is SL.
Strona 13 z 41
Key steps of the proof:
Powrót
• notice that | · |F is absolutely continuous on each set
{x ∈ [a, b] : F 0(x) exists and F 0(x) ≤ n},
n ∈ N;
Pełny ekran
so it’s enough to prove
• | · |F is absolutely continuous =⇒ F is almost everywhere differentiable.
Zamknij
Koniec
Strona główna
Strona tytułowa
LEMMA.
Bongiorno & Di Piazza & Skvortsov 1995.
| · |F is absolutely continuous =⇒ F is VBG∗, so almost everywhere differentiable.
Spis treści
A function F : [a, b] → R is said to be VBG∗ if [a, b] =
each n,
k
X
ωF (Ii) < Mn
S∞
n=1
En, where, for
JJ
II
J
I
i=1
for any collection {Ii}ki=1 of nonoverlapping intervals with both endpoints in En.
Strona 14 z 41
Powrót
LEMMA.
Every VBG∗-function is almost everywhere differentiable.
Lusin ?
FACT. An F : [a, b] → R is VBG∗-function iff | · |F is σ-finite on a co-countable
subset of [a, b].
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LEMMA.
Bongiorno & Di Piazza & Skvortsov 1995.
F is SL =⇒ F is VBG∗, so almost everywhere differentiable.
Spis treści
A function F : [a, b] → R is said to be VBG∗ if [a, b] =
each n,
k
X
ωF (Ii) < Mn
S∞
n=1
En, where, for
JJ
II
J
I
i=1
for any collection {Ii}ki=1 of nonoverlapping intervals with both endpoints in En.
Strona 15 z 41
Powrót
LEMMA.
Every VBG∗-function is almost everywhere differentiable.
Lusin ?
FACT. An F : [a, b] → R is VBG∗-function iff | · |F is σ-finite on a co-countable
subset of [a, b].
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LEMMA.
Bongiorno & Di Piazza & Skvortsov 1995.
F is SL =⇒ F is VBG∗, so almost everywhere differentiable.
Spis treści
A function F : [a, b] → R is said to be VBG∗ if [a, b] =
each n,
k
X
ωF (Ii) < Mn
S∞
n=1
En, where, for
JJ
II
J
I
i=1
for any collection {Ii}ki=1 of nonoverlapping intervals with both endpoints in En.
Strona 16 z 41
Powrót
LEMMA.
Every VBG∗-function is almost everywhere differentiable.
Lusin ?
FACT. An F : [a, b] → R is VBG∗-function iff | · |F is σ-finite on a co-countable
subset of [a, b].
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Strona tytułowa
APPROXIMATE KURZWEIL–HENSTOCK INTEGRAL
Spis treści
Let S ⊂ R. We say a tagged interval ([y, z], x) is S-fine if y, z ∈ S.
Let C = {Sx : x ∈ R} be a collection of sets. We say a division
C-fine if for each i, (Ii, xi) is Sxi -fine.
{(Ii, xi)}ki=1
JJ
II
J
I
is
Strona 17 z 41
Let a set A ⊂ R be measurable, x ∈ R. The density of A at x is, if it exists, the
value
|A ∩ [x − h, x + h]|
d(A, x) = lim
.
h→0
2h
Let ∆(x), x ∈ R, be the collection of measurable sets E ⊂ R with d(E, x) = 1.
Powrót
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Strona tytułowa
APPROXIMATE KURZWEIL–HENSTOCK INTEGRAL
Spis treści
Let S ⊂ R. We say a tagged interval ([y, z], x) is S-fine if y, z ∈ S.
Let C = {Sx : x ∈ R} be a collection of sets. We say a division
C-fine if for each i, (Ii, xi) is Sxi -fine.
{(Ii, xi)}ki=1
JJ
II
J
I
is
Strona 18 z 41
Let a set A ⊂ R be measurable, x ∈ R. The density of A at x is, if it exists, the
value
|A ∩ [x − h, x + h]|
d(A, x) = lim
.
h→0
2h
Let ∆(x), x ∈ R, be the collection of measurable sets E ⊂ R with d(E, x) = 1.
Powrót
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FACT.
Russell Gordon 1990?
For any collection C = {Sx ∈ ∆(x) : x ∈ R}, there is a C-fine partition of any
[a, b] ⊂ R.
DEFINITION.
Gordon 1990
We call a function f : [a, b] → R, AH-integrable, with the integral I =
Rb
(AH) a f ∈ R, if for each ε > 0 there exists a collection C = {Sx ∈ ∆(x) :
x ∈ R}, such that for every C-fine partition {(Ii, xi)}ki=1 of [a, b],
k
X
f (xi)|Ii| − I < ε.
i=1
Z
F (x) = (AH)
Spis treści
JJ
II
J
I
Strona 19 z 41
Powrót
Pełny ekran
x
f
a
How to characterize F ?
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Koniec
Strona główna
Strona tytułowa
FACT.
Russell Gordon 1990?
For any collection C = {Sx ∈ ∆(x) : x ∈ R}, there is a C-fine partition of any
[a, b] ⊂ R.
DEFINITION.
Gordon 1990
We call a function f : [a, b] → R, AH-integrable, with the integral I =
Rb
(AH) a f ∈ R, if for each ε > 0 there exists a collection C = {Sx ∈ ∆(x) :
x ∈ R}, such that for every C-fine partition {(Ii, xi)}ki=1 of [a, b],
k
X
f (xi)|Ii| − I < ε.
i=1
Z
F (x) = (AH)
Spis treści
JJ
II
J
I
Strona 20 z 41
Powrót
Pełny ekran
x
f
a
How to characterize F ?
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APPROXIMATE VARIATIONAL MEASURE
Strona główna
Strona tytułowa
Let F : [a, b] → R.
Spis treści
|E|ap
F stands for the approximate variational measure of E ⊂ [a, b] induced by F :
|E|ap
F = inf sup
C
P
k
X
|∆F (Ii)|,
JJ
II
J
I
i=1
where sup runs over all C-fine divisions P = {(Ii, xi)}ki=1 anchored in E, inf over
all collections C = {Sx ∈ ∆(x) : x ∈ R}.
Strona 21 z 41
Powrót
We say F is ASL (after Approximate Strong Lusin Condition) if |E|ap
F = 0 for
every nullset E ⊂ [a, b].
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APPROXIMATE VARIATIONAL MEASURE
Strona główna
Strona tytułowa
Let F : [a, b] → R.
Spis treści
|E|ap
F stands for the approximate variational measure of E ⊂ [a, b] induced by F :
|E|ap
F = inf sup
C
P
k
X
|∆F (Ii)|,
JJ
II
J
I
i=1
where sup runs over all C-fine divisions P = {(Ii, xi)}ki=1 anchored in E, inf over
all collections C = {Sx ∈ ∆(x) : x ∈ R}.
Strona 22 z 41
Powrót
We say F is ASL (after Approximate Strong Lusin Condition) if |E|ap
F = 0 for
every nullset E ⊂ [a, b].
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APPROXIMATE VARIATIONAL MEASURE
Spis treści
Ene 1998
THEOREM.
Let F : [a, b] → R. TAE:
0
❶ F is an indefinite AH-integral (of its approximate derivative Fap
),
JJ
II
J
I
❷ F is ASL.
Strona 23 z 41
Key steps of the proof:
• notice that
| · |ap
F
Powrót
is absolutely continuous on each set
0
{x ∈ [a, b] : F 0(x) exists and Fap
(x) ≤ n},
n ∈ N;
Pełny ekran
so it’s enough to prove
• | · |ap
F is absolutely continuous =⇒ F is almost everywhere approximately differentiable.
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Strona główna
Strona tytułowa
APPROXIMATE VARIATIONAL MEASURE
Spis treści
Ene 1998
THEOREM.
Let F : [a, b] → R. TAE:
0
❶ F is an indefinite AH-integral (of its approximate derivative Fap
),
JJ
II
J
I
❷ F is ASL.
Strona 24 z 41
Key steps of the proof:
• notice that
| · |ap
F
Powrót
is absolutely continuous on each set
0
{x ∈ [a, b] : F 0(x) exists and Fap
(x) ≤ n},
n ∈ N;
Pełny ekran
so it’s enough to prove
• | · |ap
F is absolutely continuous =⇒ F is almost everywhere approximately differentiable.
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Strona główna
Strona tytułowa
APPROXIMATE VARIATIONAL MEASURE AND VBG
Spis treści
To this aim, one proves
• F is ASL implies F is measurable;
JJ
II
J
I
• F is ASL =⇒ F has VBG property on each nullset;
• LEMMA.
Ene 1998
A measurable F is VBG iff it is VBG on every Lebesgue nullset E ⊂ [a, b].
• LEMMA.
Denjoy–Khintchine 1916
A measurable VBG function is a.e. approximately differentiable.
Strona 25 z 41
Powrót
A function F : [a, b] → R is said to be VBG if [a, b] =
S∞
n=1
En, where, for each n,
Pełny ekran
k
X
∆F (Ii) < Mn
i=1
Zamknij
for any collection {Ii}ki=1 of nonoverlapping intervals with both endpoints in En.
Koniec
Strona główna
Strona tytułowa
APPROXIMATE VARIATIONAL MEASURE AND VBG
Spis treści
To this aim, one proves
• F is ASL implies F is measurable;
JJ
II
J
I
• F is ASL =⇒ F has VBG property on each nullset;
• LEMMA.
Ene 1998
A measurable F is VBG iff it is VBG on every Lebesgue nullset E ⊂ [a, b].
• LEMMA.
Denjoy–Khintchine 1916
A measurable VBG function is a.e. approximately differentiable.
Strona 26 z 41
Powrót
A function F : [a, b] → R is said to be VBG if [a, b] =
S∞
n=1
En, where, for each n,
Pełny ekran
k
X
∆F (Ii) < Mn
i=1
Zamknij
for any collection {Ii}ki=1 of nonoverlapping intervals with both endpoints in En.
Koniec
Strona główna
Strona tytułowa
APPROXIMATE VARIATIONAL MEASURE AND VBG
Spis treści
To this aim, one proves
• F is ASL implies F is measurable;
JJ
II
J
I
• F is ASL =⇒ F has VBG property on each nullset;
• LEMMA.
Ene 1998
A measurable F is VBG iff it is VBG on every Lebesgue nullset E ⊂ [a, b].
• LEMMA.
Denjoy–Khintchine 1916
A measurable VBG function is a.e. approximately differentiable.
Strona 27 z 41
Powrót
A function F : [a, b] → R is said to be VBG if [a, b] =
S∞
n=1
En, where, for each n,
Pełny ekran
k
X
∆F (Ii) < Mn
i=1
Zamknij
for any collection {Ii}ki=1 of nonoverlapping intervals with both endpoints in En.
Koniec
Strona główna
Strona tytułowa
APPROXIMATE VARIATIONAL MEASURE AND VBG
Spis treści
To this aim, one proves
• F is ASL implies F is measurable;
JJ
II
J
I
• F is ASL =⇒ F has VBG property on each nullset;
• LEMMA.
Ene 1998
A measurable F is VBG iff it is VBG on every Lebesgue nullset E ⊂ [a, b].
• LEMMA.
Denjoy–Khintchine 1916
A measurable VBG function is a.e. approximately differentiable.
Strona 28 z 41
Powrót
A function F : [a, b] → R is said to be VBG if [a, b] =
S∞
n=1
En, where, for each n,
Pełny ekran
k
X
∆F (Ii) < Mn
i=1
Zamknij
for any collection {Ii}ki=1 of nonoverlapping intervals with both endpoints in En.
Koniec
Strona główna
Strona tytułowa
APPROXIMATE VARIATIONAL MEASURE AND VBG
Spis treści
To this aim, one proves
• F is ASL implies F is measurable;
JJ
II
J
I
• F is ASL =⇒ F has VBG property on each nullset;
• LEMMA.
Ene 1998
A measurable F is VBG iff it is VBG on every Lebesgue nullset E ⊂ [a, b].
• LEMMA.
Denjoy–Khintchine 1916
A measurable VBG function is a.e. approximately differentiable.
Strona 29 z 41
Powrót
A function F : [a, b] → R is said to be VBG if [a, b] =
S∞
n=1
En, where, for each n,
Pełny ekran
k
X
∆F (Ii) < Mn
i=1
Zamknij
for any collection {Ii}ki=1 of nonoverlapping intervals with both endpoints in En.
Koniec
Strona główna
PROOF OF ENE’S LEMMA
V.A.Skvortsov & PS
Strona tytułowa
Spis treści
Take P = [a, b] \
∞
S
En, |P | = 0, En = cl En, F En – continuous.
n=1
F ∈
/ VBG[a, b] implies F ∈
/ VBG(En) for some n.
D — the set of all x ∈ En such that F is VBG on no I ∩ En, I 3 x an open
interval.
JJ
II
J
I
Strona 30 z 41
Powrót
D0 — co-countable subset of D without unilaterally isolated points
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(1)
Pick I1 , . . . , Im(1)1 , m1 ≥ 2, with endpoints in D0, so that
Strona tytułowa
m1
X
∆F Ii(1) ≥ 1.
Spis treści
i=1
0
In D ∩
Sm1
(1)
I
i
i=1
pick intervals
(2)
I1 , . . . , Im(2)2
JJ
II
J
I
0
with endpoints in D , so that
(1)
(2)
① each of Ii , i = 1, . . . , m1 , contains at least two of Ij , j = 1, . . . , m2 ,
(1)
(2)
② both endpoints of every Ii , i = 1, . . . , m1 , are endpoints of some Ij , j =
1, . . . , m2;
m2
X
(2) 1
Ii < ;
③
2
i=1
④ for every i = 1, . . . , m1 ,
Strona 31 z 41
Powrót
X
∆F Ij(2) ≥ 2.
(2)
(2)
j:Ij ⊂Ii
Pełny ekran
And so on. With a category argument one can show that on the nullset
N =D∩
mk
∞ [
\
Zamknij
(k)
Ii
k=1 i=1
F is not VBG.
Koniec
Strona główna
(1)
Pick I1 , . . . , Im(1)1 , m1 ≥ 2, with endpoints in D0, so that
Strona tytułowa
m1
X
∆F Ii(1) ≥ 1.
Spis treści
i=1
0
In D ∩
Sm1
(1)
I
i
i=1
pick intervals
(2)
I1 , . . . , Im(2)2
JJ
II
J
I
0
with endpoints in D , so that
(1)
(2)
① each of Ii , i = 1, . . . , m1 , contains at least two of Ij , j = 1, . . . , m2 ,
(1)
(2)
② both endpoints of every Ii , i = 1, . . . , m1 , are endpoints of some Ij , j =
1, . . . , m2;
m2
X
(2) 1
Ii < ;
③
2
i=1
④ for every i = 1, . . . , m1 ,
Strona 32 z 41
Powrót
X
∆F Ij(2) ≥ 2.
(2)
(2)
j:Ij ⊂Ii
Pełny ekran
And so on. With a category argument one can show that on the nullset
N =D∩
mk
∞ [
\
Zamknij
(k)
Ii
k=1 i=1
F is not VBG.
Koniec
Strona główna
(1)
Pick I1 , . . . , Im(1)1 , m1 ≥ 2, with endpoints in D0, so that
Strona tytułowa
m1
X
∆F Ii(1) ≥ 1.
Spis treści
i=1
0
In D ∩
Sm1
(1)
I
i
i=1
pick intervals
(2)
I1 , . . . , Im(2)2
JJ
II
J
I
0
with endpoints in D , so that
(1)
(2)
① each of Ii , i = 1, . . . , m1 , contains at least two of Ij , j = 1, . . . , m2 ,
(1)
(2)
② both endpoints of every Ii , i = 1, . . . , m1 , are endpoints of some Ij , j =
1, . . . , m2;
m2
X
(2) 1
Ii < ;
③
2
i=1
④ for every i = 1, . . . , m1 ,
Strona 33 z 41
Powrót
X
∆F Ij(2) ≥ 2.
(2)
(2)
j:Ij ⊂Ii
Pełny ekran
And so on. With a category argument one can show that on the nullset
N =D∩
mk
∞ [
\
Zamknij
(k)
Ii
k=1 i=1
F is not VBG.
Koniec
Strona główna
MEASURABILITY IS ESSENTIAL
VAS & PS
Strona tytułowa
Spis treści
[0, 1] = {pα }α<Ω
JJ
II
J
I
{Gα }α<Ω — all Gδ null subsets of [0, 1]
Put H0 = G0 and for each α < Ω take an α̃ < Ω such that
[
Gα̃ ⊃
Hβ ∪ Gα
Strona 34 z 41
β<α
and
Gα̃ \
[
Powrót
Hβ
is uncountable.
(1)
β<α
Pełny ekran
Define Hα = Gα̃ .
Zamknij
Koniec
Strona główna
MEASURABILITY IS ESSENTIAL
VAS & PS
Strona tytułowa
Spis treści
[0, 1] = {pα }α<Ω
JJ
II
J
I
{Gα }α<Ω — all Gδ null subsets of [0, 1]
Put H0 = G0 and for each α < Ω take an α̃ < Ω such that
[
Gα̃ ⊃
Hβ ∪ Gα
Strona 35 z 41
β<α
and
Gα̃ \
[
Powrót
Hβ
is uncountable.
(2)
β<α
Pełny ekran
Define Hα = Gα̃ .
Zamknij
Koniec
Strona główna
Strona tytułowa
{Hα }α<Ω is ascending,
[
α<Ω
[
Hα =
Gα = [0, 1].
α<Ω
Put
F (x) = pα at x ∈ Hα \
Spis treści
[
JJ
II
J
I
Hβ .
β<α
F is VBG on each nullset D ⊂ [0, 1]:
D ⊂ Gα ⊂ Hα for some α < Ω. Thus F (D) ⊂ {pβ }β6α .
Strona 36 z 41
Powrót
Due to (2), F takes upon each x ∈ [0, 1] as a value uncountably many times.
Pełny ekran
Zamknij
Koniec
Strona główna
Strona tytułowa
{Hα }α<Ω is ascending,
[
α<Ω
[
Hα =
Gα = [0, 1].
α<Ω
Put
F (x) = pα at x ∈ Hα \
Spis treści
[
JJ
II
J
I
Hβ .
β<α
F is VBG on each nullset D ⊂ [0, 1]:
D ⊂ Gα ⊂ Hα for some α < Ω. Thus F (D) ⊂ {pβ }β6α .
Strona 37 z 41
Powrót
Due to (2), F takes upon each x ∈ [0, 1] as a value uncountably many times.
Pełny ekran
Zamknij
Koniec
Strona główna
Suppose [0, 1] =
∞
[
En and F is VB on each En.
Strona tytułowa
n=1
In = {pα : En ∩ F −1(pα ) is infinite}, n ∈ N.
|In| > 0 for some n, since [0, 1] =
∞
[
In
Spis treści
JJ
II
J
I
n=1
Consider the indicatrix function I of F En. By the Banach indicatrix theorem,
Z∞
I 6 V (F En) < ∞.
Strona 38 z 41
−∞
On the other hand, I(y) = ∞ for each y ∈ In, whence
Powrót
Z∞
I > ∞ · |In| = ∞,
Pełny ekran
−∞
a contradiction.
Zamknij
Koniec
Strona główna
Suppose [0, 1] =
∞
[
En and F is VB on each En.
Strona tytułowa
n=1
In = {pα : En ∩ F −1(pα ) is infinite}, n ∈ N.
|In| > 0 for some n, since [0, 1] =
∞
[
In
Spis treści
JJ
II
J
I
n=1
Consider the indicatrix function I of F En. By the Banach indicatrix theorem,
Z∞
I 6 V (F En) < ∞.
Strona 39 z 41
−∞
On the other hand, I(y) = ∞ for each y ∈ In, whence
Powrót
Z∞
I > ∞ · |In| = ∞,
Pełny ekran
−∞
a contradiction.
Zamknij
Koniec
Strona główna
SOME GENERALIZATIONS RELATED TO MEASURABILITY
Strona tytułowa
Extra assumptions on {∆(x)}x∈R:
Spis treści
❶ if S ∈ ∆(x) then there exists a measurable set R ⊂ S such that
d(R, x) > 0;
❷ if S ∈ ∆(x) and d(R, x) = 1, R 3 x, then
JJ
II
J
I
S ∩ R ∈ ∆(x).
Strona 40 z 41
THEOREM.
VAS & PS 2012
∆
Let F : [a, b] → R. Assume | · |F is σ-finite on each nullset. Then F : R → R is
measurable.
THEOREM.
VAS & PS 2002
ap
If F : R → R is measurable, then | · |F is σ-finite on each nullset implies it
is σ-finite on [a, b]. In turn this implies F is almost everywhere approximately
differentiable.
COROLLARY.
VAS & PS 2012
∆ap
ap
Assume VF is σ-finite on each nullset. Then VF is σ-finite and F is approximately differentiable a.e.
Powrót
Pełny ekran
Zamknij
Koniec
Strona główna
SOME GENERALIZATIONS RELATED TO MEASURABILITY
Strona tytułowa
Extra assumptions on {∆(x)}x∈R:
Spis treści
❶ if S ∈ ∆(x) then there exists a measurable set R ⊂ S such that
d(R, x) > 0;
❷ if S ∈ ∆(x) and d(R, x) = 1, R 3 x, then
JJ
II
J
I
S ∩ R ∈ ∆(x).
Strona 41 z 41
THEOREM.
VAS & PS 2012
∆
Let F : [a, b] → R. Assume | · |F is σ-finite on each nullset. Then F : R → R is
measurable.
THEOREM.
VAS & PS 2002
ap
If F : R → R is measurable, then | · |F is σ-finite on each nullset implies it
is σ-finite on [a, b]. In turn this implies F is almost everywhere approximately
differentiable.
COROLLARY.
VAS & PS 2012
∆ap
ap
Assume VF is σ-finite on each nullset. Then VF is σ-finite and F is approximately differentiable a.e.
Powrót
Pełny ekran
Zamknij
Koniec