Variational measures generated by functions and associated with - - PDF document

Variational measures generated by functions and associated with local systems of sets

Luisa Di Piazza University of Palermo Department of Mathematics ITALY

In recent years several authors have been in- terest in the variational measures generated by functions. Roughly speaking, given a real function f on R it is possible to construct, using suitable families

• f intervals, appropriate measures µf which carry

variational informations about f. In case the family of intervals is the full interval basis, perhaps the nicest application of these measures is the following claim Let f : [a, b] → R. Then the identity f(x) − f(a) =

x

a f′(t)dt

holds in the sense of the Lebesgue integral (resp.

• f the Kurzweil-Henstock integral) if and only

if the measure µf is finite and absolutely con- tinuous (resp. absolutely continuous) with re- spect to the Lebesgue measure. Aim of this talk is to consider properties of vari- ational measures generated by functions and as- sociated with local systems of sets.

1

• 1. Local systems

Following B.S. Thomson (1985) we call local system any family S = {S(x) : x ∈ R} of collec- tions of sets S(x) such that: (i) {x} / ∈ S(x), for all x ∈ R; (ii) if s ∈ S(x), then x ∈ s; (iii) if s ∈ S(x) and δ > 0, then s ∩ (x − δ, x + δ) ∈ S(x). (iv) if s1 ∈ S(x) and s1 ⊆ s2, then s2 ∈ S(x); A local system S is said to be filtering if for each x ∈ R and each s1, s2 ∈ S(x) it is s1 ∩ s2 ∈ S(x).

2

Examples 1) The interval local system: for each x ∈ R, S(x) contains all the neighborhoods of x. 2) The approximate local system: for each x ∈

R, the family S(x) contains all the sets E

such that x ∈ E and x is a density point of E. 3) Systems generated by paths (or path-systems): for any x ∈ R there exists a set Ex ⊆ R (called the path at x) such that a) x ∈ Ex, b) x is a point of accumulation of Ex, and S(x) is the filter generated by {Ex ∩ (x − η, x + η) : η > 0}.

3

4) Path P-adic system: let P = {pj}∞

j=0 be a

sequence of integers, with pj > 1 for j = 0, 1, .... We set m0 = 1, mk = p0p1....pk−1, for k ≥ 1. For fixed k = 0, 1, ... the intervals

• r

mk , r + 1 mk

• = I(k)

r

, r ∈ Z are called P-adic intervals of rank k. For x ∈ R we denote by P−(x) (resp. by P+(x)) the sequence of all left end-points (resp. all right end-points) of the P-adic intervals containing x. The set Ex = {x} ∪ P−(x) ∪ P+(x) is the P-adic path at x. We denote by P the path–system generated by the P-adic paths. In case P = {2, 2, 2, ....} we obtain the more familiar path dyadic system.

4

• 2. Choise and partition

Given a local system S and E ⊆ R we call S- choice (or simply choice) on E any function γ : E → 2R such that γ(x) ∈ S(x). Given a choice γ we set βγ = {([u, v], x) : x = u, v ∈ γ(x)

• r

x = v, u ∈ γ(x); x ∈ R} , and for a set E ⊂ R we put βγ[E] = {([u, v], x) ∈ βγ : x ∈ E}. A finite subset π of βγ[E] is called a βγ -partition

• n E if for distinct elements (I1, x1) and (I2, x2)

in π, the intervals I1 and I2 are nonoverlapping. If

(I,x)∈π I = E, π is called a βγ -partition of E.

5

• 3. The Variational S-Measure

Let S be a filtering local system and let F : E →

• R. Given an S-choice γ on E we set

V ar(βγ, F, E) = sup

π⊂βγ[E]

• (I,x)∈π

|F(I)| , where if I = [u, v] we use the notation F(I) = F(v) − F(u). We also set V S

F (E) = inf γ V ar(βγ, F, E),

where “inf” is taken over all choices γ on E. V S

F is a metric outer measure, called the varia-

tional measure generated by F with respect to the system S.

6

• 4. S-Limit and S-Derivative

Let S be a local system and let F be a real function on R. We say that (S) lim

t→x F(t) = c

if for every ε > 0 the set {t : t = x or |F(t)−c| < ε} ∈ S(x). When c = F(x) the function F is said to be S-continuous at the point x. Notice that if the local system is filtering, then the S-limit is unique.

7

The lower S derivative of (resp. the upper S derivative of) the function F at a point x is de- fined by DSF(x) = sup

s∈S(x)

inf

• F(y) − F(x)

y − x : y ∈ s, y = x

• ,

(resp. DSF(x) = inf

s∈S(x) sup

• F(y) − F(x)

y − x : y ∈ s, y = x

• ).

If DSF(x) = DSF(x) and this value is finite, we say that F is S-differentiable at x and we set DSF(x) = DSF(x) = DSF(x).

8

Proposition 1. Let F be an S-continuos func-

• tion. If the variational measure V S

F is σ-finite on

E ∈ L, then the extreme S-derivatives are finite almost everywhere on E. In the following we say that an outer measure µ is absolutely continuous if µ is absolutely conti- nuous with respect to the Lebesgue measure Λ (br. µ ≪ Λ): i.e. Λ(N) = 0 ⇒ µ(N) = 0. Proposition 2. If S is one of the following local systems:

• the interval local system,
• the approximate local system,
• the path-dyadic system, or more in general a

path P-adic system then: V S

F ≪Λ ⇒ V S F is σ-finite.

• (B. Bongiorno, V. Skvortsov, L. DP 1995-96)
• (V. Skvortsov, P. Sworowski, 2002)
• (B. Bongiorno, V. Skvortsov, L. DP 2002,
• D. Bongiorno, V. Skvortsov, L. DP 2006)

9

• 5. The Ward property

We say that a local system S possesses the Ward property whenever each function is S- differentiable almost everywhere on the set of all points at which at least one of its extreme S-derivatives is finite. Theorem 1. Let S be or the interval local system or the approximate local system. If the variational measure V S

F is absolutely continuous

• n X ∈ L, then the function F is S-differentiable

almost everywhere on X. (B. Bongiorno, V. Skvortsov and L. DP 1995- 96, V. Skvortsov and P. Sworowski, 2002)

10

Theorem 2. (see [11]) Each P-adic path sys- tem defined by a bounded sequence P={pj}∞

j=0,

possesses the Ward property. Let us remark that, in case the sequence P={pj}∞

j=0

is unbounded, then the Ward property may fail to be true (see D. Bongiorno, V. Skvortsov, L. DP 2006). Theorem 3. Let P = {pj}∞

j=0 be a bounded

• sequence. If the variational measure V P

F

is ab- solutely continuous on X ∈ L, then the function F is P-differentiable almost everywhere on X.

11

• 6. S-ACG Functions

A function F is said to be S-AC on a set E ⊂ R if for any ε > 0 there exists δ > 0 and an S-choice γ on E such that

• (I,x)∈π

|F(I)| < ε, for any partition π ∈ βγ[E] with

• (I,x)∈π

|I| < δ. F is said to be S-ACG on E if E =

n En, and

F is S-AC on En, for each n.

12

Results

• 1. For any local system S we have

S-ACG ⊆ {F : VS

F ≪ Λ},

(V . Ene, 1995);

• 2. If S is the interval local system, then we have

S-ACG = {F : VS

F ≪ Λ},

(R. Gordon, 1994);

• 3. If S is the approximate local system, then

we have S-ACG = {F : VS

F ≪ Λ},

(V . Ene, 1998).

• 4. If S is a path P-adic system, then we have

S-ACG = {F : VS

F ≪ Λ},

(D. Bongiorno − V . Skvortsov − L.DP, 2006).

13

• 7. Applications to the S-integral

Let S be a local system which is filtering and satisfying the partitioning property (i.e. for any S-choice γ there exists a βγ-partition of any in- terval of R). Definition A function f : [a, b] → R is said to be S-integrable on [a, b], with integral A, if for every ε > 0 there exists a choice γ on [a, b] such that

• (I,x)∈π

f(x)|I| − A

• < ε ,

for any partition π ⊂ βγ of [a, b]. We write A = (S)

b

a f.

14

Properties.

• If a function f is S-integrable on [a, b], then it

is also S-integrable on each subinterval of [a, b]. The function F(x) := (S)

x

a f is called the in-

definite S-integral.

• A function F is the indefinite S-integral of a

function f on [a, b] if and only if i) V S

F ≪ Λ on [a, b] and

ii) F is S-differentiable a.e. with DSF(x) = f(x) a.e. on [a, b];

• r if and only if

j) F is SACG on [a, b] and jj) F is S-differentiable a.e. with DSF(x) = f(x) a.e. on [a, b].

15

Wether for general local systems S the as- sumption of S-differentiability can be dropped in both the ”if“ parts of previous characterizations

• f the S-indefinite integral, is an open question.

As applications of the results in sections 5-6 we get that for some particular local systems the answer is positive. Theorem 4. Let S be one of the following local systems:

• the interval local system
• the approximate local system
• a path P-adic system defined by a bounded

sequence P. Then F is the indefinite S-integral of a function

• n [a, b] if and only if V S

F ≪ Λ, on [a, b]; or if and

• nly if F is S−ACG on [a, b].

16

References 1 Bongiorno B., Di Piazza L., Skvortsov V.A., A new full descriptive characterization of the Denjoy-Perron integral, Real Anal. Exchange. 21, No. 2 (1995-96), 656–663. 2 Bongiorno B., Di Piazza L., Skvortsov V.A., On variational measures related to some bases,

• J. Math. Anal. Appl. 250 (2000), 533–547.

3 Bongiorno B., Di Piazza L., Skvortsov V.A., On dyadic integrals and some other integrals associated with local systems, J. Math. Anal. Appl. 271 (2002), 506–524. 4 Bongiorno B., Di Piazza L., Skvortsov V.A., The Ward property for a P-adic basis and the P-adic integral.

• J. Math. Anal. Appl.

285 (2003), 578–592. 5 Bongiorno D., Di Piazza L., Skvortsov V.A., Variational measures related to local systems

17

and the Ward property of P-adic bases, Czechoslo- vak Math. J. 56 131 (2006), 559–578. 6 Di Piazza L., Variational measures in the theory of the integration in Rm, Czechoslo- vak Math. J. 51 No 1 (2001), 95–110. 7 Ene V., Thomson’s variational measures, Real

• Anal. Exchange 24 No. 2, (1998/99), 523–

566. 8 Skvortsov V. A., Sworowski P., On a vari- ational measure defined by an approximate differential basis, (Russian) Vestnik Moskov.

• Univ. Ser. I Mat. Mekh. 2002, , no. 1, 54–

57, 72; translation in Moscow Univ. Math.

• Bull. 57 (2002), no. 1, 37–40

9 Thomson B.S., Some property of variational measures, Real Anal. Exchange 24, No. 2 (1998/99), 845–854. 10 Thomson B.S., Real functions, Lecture Notes in Math., Vol. 1170, Springer–Verlag, 1980.

11 Tulone F., On the Ward Theorem for P- adic path bases associated with a bounded sequence P . Math. Bohem, 129 (2004),

• no. 3, 313–323.