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 of intervals, appropriate measures µ f 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 � x a f ′ ( t ) dt f ( x ) − f ( a ) = holds in the sense of the Lebesgue integral (resp. of the Kurzweil-Henstock integral) if and only if the measure µ f 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 s 1 ∈ S ( x ) and s 1 ⊆ s 2 , then s 2 ∈ S ( x ); A local system S is said to be filtering if for each x ∈ R and each s 1 , s 2 ∈ S ( x ) it is s 1 ∩ s 2 ∈ 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 E x ⊆ R (called the path at x ) such that a) x ∈ E x , b) x is a point of accumulation of E x , and S ( x ) is the filter generated by { E x ∩ ( x − η, x + η ) : η > 0 } . 3

Path P -adic system: let P = { p j } ∞ 4) j =0 be a sequence of integers, with p j > 1 for j = 0 , 1 , ... . We set m 0 = 1, m k = p 0 p 1 ....p k − 1 , for k ≥ 1. For fixed k = 0 , 1 , ... the intervals � � , r + 1 r = I ( k ) , r ∈ Z r m k m k 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 E x = { 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 → 2 R such that γ ( x ) ∈ S ( x ). Given a choice γ we set β γ = { ([ u, v ] , x ) : x = u, v ∈ γ ( x ) or x = v, u ∈ γ ( x ); x ∈ R } , and for a set E ⊂ R we put β γ [ E ] = { ([ u, v ] , x ) ∈ β γ : x ∈ E } . A finite subset π of β γ [ E ] is called a β γ -partition on E if for distinct elements ( I 1 , x 1 ) and ( I 2 , x 2 ) in π , the intervals I 1 and I 2 are nonoverlapping. If � ( I,x ) ∈ π I = E , π is called a β γ -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 ( β γ , F, E ) = sup | F ( I ) | , π ⊂ β γ [ E ] ( I,x ) ∈ π 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 ( β γ , 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 � � F ( y ) − F ( x ) D S F ( x ) = sup inf : y ∈ s, y � = x , y − x s ∈ S ( x ) (resp. � � F ( y ) − F ( x ) D S F ( x ) = s ∈ S ( x ) sup inf : y ∈ s, y � = x ) . y − x If D S F ( x ) = D S F ( x ) and this value is finite, we say that F is S -differentiable at x and we set D S F ( x ) = D S F ( x ) = D S F ( 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 on 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 = { p j } ∞ j =0 , possesses the Ward property. Let us remark that, in case the sequence P = { p j } ∞ j =0 is unbounded, then the Ward property may fail to be true (see D. Bongiorno, V. Skvortsov, L. DP 2006). Let P = { p j } ∞ Theorem 3. j =0 be a bounded sequence. If the variational measure V P is ab- F 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 � | F ( I ) | < ε, ( I,x ) ∈ π for any partition π ∈ β γ [ E ] with � | I | < δ. ( I,x ) ∈ π F is said to be S -ACG on E if E = � n E n , and F is S -AC on E n , for each n . 12

Results 1. For any local system S we have S -ACG ⊆ { F : V S F ≪ Λ } , ( V . Ene , 1995 ); 2. If S is the interval local system, then we have S -ACG = { F : V S ( R . Gordon , 1994 ); F ≪ Λ } , 3. If S is the approximate local system, then we have S -ACG = { F : V S F ≪ Λ } , ( V . Ene , 1998 ) . 4. If S is a path P -adic system, then we have S -ACG = { F : V S 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 β γ -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 � � � � � � � f ( x ) | I | − A < ε , � � � � � � ( I,x ) ∈ π � � for any partition π ⊂ β γ of [ a, b ]. We write � b A = ( S ) 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 ]. � x The function F ( x ) := ( S ) 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 D S F ( x ) = f ( x ) a.e. on [ a, b ]; or if and only if j ) F is S ACG on [ a, b ] and jj ) F is S -differentiable a.e. with D S F ( 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 of 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 on [ a, b ] if and only if V S F ≪ Λ , on [ a, b ]; or if and only if F is S− ACG on [ a, b ]. 16