Abbas Edalat. Department of Computing. Imperial College of Science, Technology and Medicine. 180 Queen's Gate. London SW7 2BZ UK.

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1 Doman Theory and Integraton Abbas Edalat Department of Computng Imperal College of Scence, Technology and Medcne 180 Queen's Gate London SW7 2B UK Abstract We present a doman-theoretc framework for measure theory and ntegraton of bounded real-valued functons wth respect to bounded Borel measures on compact metrc spaces. The set of normalsed Borel measures of the metrc space can be embedded nto the maxmal elements of the normalsed probablstc power doman of ts upper space. Any bounded Borel measure on the compact metrc space can then be obtaned as the least upper bound of an!-chan of lnear combnatons of pont valuatons (smple valuatons) on the upper space, thus provdng a constructve setup for these measures. We use ths settng to dene a new noton of ntegral of a bounded real-valued functon wth respect to a bounded Borel measure on a compact metrc space. By usng an!-chan of smple valuatons, whose lub s the gven Borel measure, we can then obtan ncreasngly better approxmatons to the value of the ntegral, smlar to the way the Remann ntegral s obtaned n calculus by usng step functons. We show that all the basc results n the theory of Remann ntegraton can be extended n ths more general settng. Furthermore, wth ths new noton of ntegraton, the value of the ntegral, when t exsts, concdes wth the Lebesgue ntegral of the functon. An mmedate area for applcaton s n the theory of terated functon systems wth probabltes on compact metrc spaces, where we obtan a smple approxmatng sequence for the ntegral of a real-valued contnuous functon wth respect to the nvarant measure. 1 Introducton The theory of Remann ntegraton of real-valued functons was developed by Cauchy, Remann, Steltjes, and Darboux, amongst other mathematcans of the 19th century. Wth ts smple, elegant and constructve nature, t soon became, as t s today, a sold bass of calculus; t s now used n all branches of scence. The theory, however, has ts lmtatons n the followng man areas, lsted here not n any partcular order of sgncance: To appear n Theoretcal Computer Scence,

2 () It only works for ntegraton of functons dened n R n. () It can only deal wll ntegraton of functons wth respect to the Lebesgue measure,.e. the usual measure, on R n. () Unbounded functons have to be treated separately. (v) The theory lacks certan convergence propertes. For example, the pontwse lmt of a unformly bounded sequence of Remann ntegrable functons may fal to be Remann ntegrable. (v) A functon wth a `large' set of dscontnuty,.e. wth non-zero Lebesgue measure, does not have a Remann ntegral. In the early years of ths century, Lebesgue and Borel, amongst others, lad the foundaton of a new theory of ntegraton. Wth ts further development, the new theory, the so-called Lebesgue ntegraton, has become the bass of measure theory and functonal analyss. A specal case of the Lebesgue ntegral, the so-called Lebesgue-Steltjes ntegral, has also played a fundamental role n probablty theory. The underlyng bass of the Lebesgue theory s n sharp contrast to that of the Remann theory. Whereas, n the theory of Remann ntegraton, the doman of the functon s parttoned and the ntegral of the functon s approxmated by the lower and upper Darboux sums nduced by the partton, n the theory of Lebesgue ntegraton, the range of the functon s parttoned to produce smple functons whch approxmate the functon, and the ntegral s dened as the lmt of the ntegrals of these smple functons. The latter framework makes t possble to dene the ntegral of measurable functons on abstract measurable spaces, n partcular on topologcal spaces equpped wth Borel measures. Lebesgue ntegraton also enjoys very general convergence propertes, gvng rse to the complete L p -spaces. Moreover, when the Remann ntegral of a functon exsts, so does ts Lebesgue ntegral and the two values concde,.e. Lebesgue ntegraton ncludes Remann ntegraton. Nevertheless, despte these desred features, Lebesgue ntegraton s qute nvolved and much less constructve than Remann ntegraton. Consequently, Remann ntegraton remans the preferred theory wherever t s adequate n practce, n partcular n advanced calculus and n the theory of derental equatons. A number of theores have been developed to generalse the Remann ntegral whle tryng to retan ts constructve qualty. The most well-known and successful s of course the Remann-Steltjes ntegral. In more recent tmes, E. J. McShane [22] has developed a Remann-type ntegral, whch ncludes for example the Lebesgue-Steltjes ntegral, but t unfortunately falls short of the constructve features of the Remann ntegral. A new dea n measure theory on second countable locally compact Hausdor spaces was presented n [9]. It was shown that the set of normalsed Borel measures on such a space can be embedded nto the maxmal elements of the probablstc power doman of ts upper space. The mage of the embeddng conssts of all normalsed valuatons on the 2

3 upper space whch are supported n the set of maxmal elements of the upper space,.e. the sngletons of the space. Ths upper space s an!-contnuous dcpo (drected complete partal order), and t follows that ts probablstc power doman s also an!-contnuous dcpo wth a bass consstng of lnear combnatons of pont valuatons (smple valuatons) on the upper space. The mportant consequence s that any bounded Borel measure on the space can be approxmated by smple valuatons on the upper space, and we have a constructve framework for measure theory on locally compact second countable Hausdor spaces. In ths paper, we use the above doman-theoretc framework to present a novel approach n the theory of ntegraton of bounded functons wth respect to a bounded Borel measure on a compact metrc space. Instead of approxmatng the functon wth smple functons as t s done n the Lebesgue theory, we approxmate the normalsed measure wth normalsed smple valuatons on the upper space; ths provdes us wth generalsed lower and upper Darboux sums, whch we use to dene the ntegral. The ordnary theory of Remann ntegraton, as well as the Remann-Steltjes ntegraton, s precsely a partcular case of ths approach, snce any partton of, say, the closed unt nterval n fact provdes a smple normalsed valuaton on the upper space of the nterval whch gves an approxmaton to the Lebesgue measure. We therefore work n the normalsed probablstc power doman of the upper space and develop a new theory of ntegraton, called R-ntegraton, wth the followng results. R-ntegraton satses all the elementary propertes requred for a theory of ntegraton. For ntegraton wth respect to the Lebesgue measure on compact real ntervals, R-ntegraton and Remann ntegraton are equvalent. All the basc results n the theory of ordnary Remann ntegraton can be generalsed to R-ntegraton. In partcular, a functon s R-ntegrable wth respect to a bounded Borel measure on a compact metrc space t s contnuous almost everywhere. When the R-ntegral of a functon (wth respect to a bounded Borel measure on a compact metrc space) exsts so does ts Lebesgue ntegral and the two ntegrals are equal. Therefore, our theory, whch ncludes the Lebesgue-Steltjes ntegral, s a fathful and sound generalsaton of Remann ntegraton; t overcomes the lmtatons () and () mentoned above, whle retanng the constructve nature of Remann ntegraton. In practce, we are often only nterested n the ntegral of functons whch are not too dscontnuous,.e. R-ntegraton s sucent at least for bounded functons. We apply the new theory to obtan a smple approxmatng sequence for the ntegral of a real-valued almost everywhere contnuous functon wth 3

4 respect to the unque nvarant measure of an terated functon system wth probabltes on any compact metrc space. 2 A Constructve Framework for Measure Theory In ths secton, we rst revew the doman-theoretc framework for measure theory on locally compact second countable spaces whch was establshed n [9]. We wll also present some of the background results, n partcular from [17], that we need here. We wll use the standard termnology and notatons of doman theory, as for example n [18]. Gven a dcpo (D; v) and a subset A D, we let "A = fd 2 D j 9a 2 A: a v dg and "A = fd 2 D j 9a 2 A: a dg where s the way-below relaton n D. We denote the lattce of open sets of a topologcal space by. Gven a mappng f :! Y of topologcal spaces and a subset a, we denote the forward mage of a by f[a],.e. f[a] = ff(x) j x 2 ag. Fnally, for a subset a of a compact metrc space, the dameter of a s denoted by jaj. 2.1 The Upper Space Recall [25] that gven any Hausdor topologcal space, ts upper space U s the set of all non-empty compact subsets of wth the upper topology whch has basc open sets a = fc 2 U j C ag for any open set a 2. The followng propertes are easy consequences of ths denton. (See [9].) The specalsaton orderng v u of U s reverse ncluson,.e. A v u B def () 8a 2 [A a ) B a] () A B: Furthermore (U; ) s a bounded complete dcpo, n whch the least upper bound of a drected set of elements s the ntersecton of these elements and the Scott topology renes the upper topology. The sngleton map s :! U x 7! fxg embeds onto the set of maxmal elements of U. Proposton 2.1 [9] Let be a second countable locally compact Hausdor space. () The dcpo (U ; ) s!-contnuous. () The Scott topology on (U ; ) concdes wth the upper topology. () The way below relaton B C holds n (U; ) C s contaned n the nteror of B as subsets of. 4

5 (v) (U ; ) can be gven an eectve structure. From any countable bass B of consstng of relatvely compact neghbourhoods 1, we can get an order bass of U consstng of the nte unons of elements of B. Therefore, any second countable locally compact Hausdor space, can be embedded nto ts upper space U whch can be gven an eectve structure. We would lke to have a smlar embeddng for the set of bounded Borel measures on. For ths, we use the probablstc power doman of U. 2.2 The Probablstc Power Doman Recall from [7, 24, 21, 16] that a valuaton on a topologcal space Y map : Y! [0; 1) whch satses: () (a) + (b) = (a [ b) + (a \ b) () (;) = 0 () a b ) (a) (b) A contnuous valuaton [21, 17, 16] s a valuaton such that whenever A (Y ) s a drected set (wrt [ ) of open sets of Y, then ( O) = sup O2A (O): O2A For any b 2 Y, the pont valuaton based at b s the valuaton b : (Y )! [0; 1) dened by ( 1 f b 2 O b (O) = 0 otherwse: Any nte lnear combnaton n =1 r b of pont valuatons b wth constant coecents r 2 [0; 1), (1 n) s a contnuous valuaton on Y, whch we call a smple valuaton. The probablstc power doman, P Y, of a topologcal space Y conssts of the set of contnuous valuatons on Y wth (Y ) 1 and s ordered as follows: v for all open sets O of Y, (O) (O): The partal order (P Y; v) s a dcpo wth bottom n whch the lub of a drected set h 2I s gven by F =, where for O 2 (Y ) we have (O) = sup 2I (O): The probablstc power doman gves rse to a functor P : DCPO! DCPO on the category of dcpo's and contnuous s a 1 A relatvely compact subset of a topologcal space s one whose closure s compact 5

6 functons [16]. Gven a contnuous functon f : Y! between dcpo's Y and, the contnuous functon P f : P Y! P s dened by P f()(o) = (f 1 (O)). For convenence, we therefore wrte P f() = f 1. For later use we need the followng property of ths functor. Proposton 2.2 The functor P : DCPO! DCPO s locally contnuous,.e. t s Scott contnuous on homsets. Proof Let hf 2I be a drected famly of maps f : Y! n the functon space Y!. Let f = F 2I f. It s easy to see that for any open set O, we have f 1 (O) = S f 1 (O). Now let 2 P Y. Then, we get: F (P f )()(O) = (P f)()(o) = (f 1 (O)) = ( S f 1 (O)) = sup (f 1 (O)) = F ( f 1 )(O) = ( F P f )()(O): It s easy to see that, f Y s a dcpo, the map : Y! P Y b 7! b s contnuous. Furthermore, there s a nce charactersaton of the partal order on smple valuatons on a dcpo, aptly called the splttng lemma by Jones. Proposton 2.3 [16, page 84] Let Y be a dcpo. For two smple valuatons 1 = b2b r b b 2 = c2c s c c n P Y, we have: 1 v 2, for all b 2 B and all c 2 C, there exsts a nonnegatve number t b;c such that c2c and t b;c 6= 0 mples b v c. t b;c = r b b2b t b;c s c Proof The `f' part s the splttng lemma [16, page 84]. For the `only f' part, assume the condton above holds and let O 2 Y. Put A = O \ B. Then, we have: 1 (O) = P b2a r b = P P b2a bvc t b;c Therefore 1 v 2. P P P c2c\"a bvc t b;c c2c\"a s c = 2 (O): If Y s a contnuous dcpo, then there s an analogue of the splttng lemma for the way-below relaton. Frst we need the followng charactersaton of the way-below relaton. 6

7 Proposton 2.4 [19, page 46] Let = P b2b r b b be a smple valuaton and a contnuous valuaton on a contnuous dcpo. Then for all A B we have r b < ("A): b2a Proposton 2.5 Let Y be a contnuous dcpo. For two smple valuatons 1 = b2b r b b 2 = c2c s c c n P Y, we have 1 2, for all b 2 B and all c 2 C, there exsts a nonnegatve number t b;c such that c2c and t b;c 6= 0 mples b c. t b;c = r b P b2a r b = P b2a b2b t b;c < s c Proof The `only f' part s shown n [16, page 87]. For the `f' part, assume the above condton holds for 1 and 2. Let A B, then P bc t b;c P c2c\"a P bc t b;c < P c2c\"a s c = 2 ("A): It follows, by Proposton 2.4, that 1 2. The followng mportant result was establshed n [16, pages 94-98] and appears n [17]. Theorem 2.6 If Y s an (!)-contnuous dcpo then P Y s also (!)-contnuous and has a bass consstng of smple valuatons. 2.3 Extendng Valuatons to Measures We also need some results about the extensons of contnuous valuatons to Borel measures. Recall that a Borel measure on a locally compact Hausdor space s regular f for all Borel subsets B of, we have: (B) = nf f(o) j B O; O openg = sup f(k) j B K; K compactg: Any bounded measure on a -compact and locally compact Hausdor space s regular [23, page 50]. In partcular any bounded Borel measure on a second countable locally compact Hausdor space s regular [20, page 344]. Furthermore, we have: 7

8 Proposton 2.7 [9] On a locally compact second countable Hausdor space, bounded Borel measures and contnuous valuatons concde. We also recall the followng result of J. Lawson. Frst, recall that the lattce Y of open sets of a locally quas-compact sober space Y s a contnuous dstrbutve lattce and Y s n fact somorphc wth the spectrum Spec(Y ), consstng of non-unt prme elements of Y wth the hull-kernel topology. The Lawson topology on Y nduces a topology on Spec(Y ) and hence on Y whch s ner than the orgnal topology of Y. (See [13, page 252].) Proposton 2.8 [21, page 221] Any contnuous valuaton on a second countable locally quas-compact sober space Y extends unquely to a regular Borel measure on Y equpped wth the relatve Lawson topology nduced from Y. For an!-contnuous bounded complete dcpo Y, the relatve Lawson topology nduced from Y concdes wth the Lawson topology on Y whch s compact and Hausdor [1, Exercse ]. We then obtan the followng. Corollary 2.9 Any contnuous valuaton on an!-contnuous bounded complete dcpo Y extends unquely to a regular measure on Y equpped wth ts compact Lawson topology. For!-contnuous dcpo's wth bottom, whch we wll only be concerned wth n the next sectons, we can gve a more drect extenson result usng a lemma by Saheb-Djahrom as follows. Lemma 2.10 [24, page24] The lub of any!-chan h 0 of smple valuatons on a dcpo Y wth (Y ) = 1 extends unquely to a Borel measure on Y. Proposton 2.11 Any contnuous valuaton on an!-contnuous dcpo Y wth bottom extends unquely to a Borel measure on Y. Proof If (Y ) = 0, then the result s trval. Otherwse, we can assume wthout loss of generalty,.e. by a rescalng, that (Y ) = 1. By Theorem 2.6, there exsts an!-chan h 0 of smple valuatons wth F =. For each 0, let + = + (1 (Y ))?. Then, t s easy to check that h + 0 s an!-chan of smple valuatons wth (Y ) = 1 and wth lub. It follows by Lemma 2.10 that extends unquely to a Borel measure on Y. 2.4 Measure Theory va Doman Theory In [9], a sutable computatonal framework for measure theory on a locally compact Hausdor space has been establshed usng the probablstc power doman of the upper space of. We recall the man results here. Snce U s!-contnuous so s therefore P U. 8

9 Proposton 2.12 [9, Proposton 5.8] For any open set a 2, the sngleton map s :! U nduces a G subset s[a] U. Corollary 2.13 Any Borel subset B nduces a Borel subset s[b] U. For 2 P U, let be ts unque extenson to a Borel measure on U gven by Proposton 2.11 above. Let S() P U denote the set of valuatons whch are supported on the Borel set s[] of maxmal elements of U,.e. S() = f 2 P U j (U s[]) = 0g. We have 2 S() (a) = (s[a]) for all a 2. Furthermore, S() wll be a subdcpo of P U. Let M() be the set of Borel measures on whch are bounded by one (() 1). Dene a partal order on M() by v (O) (O) for all open sets O 2. Then M() wll also be a dcpo. Let M 1 () M() be the subset of normalsed measures (() = 1). Smlarly, dene P 1 U P U and S 1 () S(). Proposton 2.14 [9, Proposton 5.17] If 2 S 1 () then s a maxmal element of P U. The man result s the followng. Theorem 2.15 [9, Theorem 5.20] The dcpo's M() and S() are somorphc va the maps e : M()! S() wth e() = s 1 and j : S()! M() wth j() = s. Moreover, these maps restrct to gve an somorphsm between M 1 () and S 1 (). We can therefore dentfy M() wth S() P U. But P U has a bass consstng of smple valuatons whch can be used to provde t wth an eectve structure. Ths therefore gves us a constructve framework for bounded Borel measures on. Important Note: For convenence, we often dentfy wth e() and wrte nstead of e(). Therefore, dependng on the context, can ether be a Borel measure on or a valuaton on U whch s supported on s[]. We wll also wrte the unque extenson smply as. Example 2.16 Let = [0; 1] be the unt nterval wth the Lebesgue measure. Each partton q : 0 = x 0 < x 1 < < x j 1 < x j < x j+1 < < x N 1 < x N = 1 of [0; 1] gves rse to a smple valuaton q = N j=1 where b j = [x j 1 ; x j ] and r j = x j x j 1. 9 r j bj

10 Now consder the!-chan h q 0 of smple valuatons whch are obtaned by the sequence of parttons hq 0, where q conssts of dyadc numbers x j = j=2 for j = 0; 1; 2; 3; ; 2. In more detal, q = 2 j=1 1 2 b j wth b j = [ j 1 2 ; j 2 ]: For any open nterval a [0; 1], the number q (a) s the largest dstance between dyadc numbers x j whch are contaned n a. For an arbtrary open set, the contrbutons from ndvdual connected components (ntervals) add up. It s easy to see that q v for all 0. Furthermore, we have: Proposton 2.17 The valuaton = F 0 q j() s the Lebesgue measure on [0; 1]. s supported n s[] and Proof Let ha k k2j n, n 1, be the collecton of all open balls n [0; 1] wth radus at most 1=n, and put [ O n = a k : k2jn Then, ho n n1 s a decreasng sequence of open sets n U[0; 1] and s[[0; 1]] = T n1 O n. But for each n 1, q (O n ) = 1 f 1=2 < 1=n,.e. f s large enough. Hence, (O n ) = sup q (O n ) = 1 for all n 1. Therefore, (s[[0; 1]]) = nf n1 (O n ) = 1 showng that 2 S 1 ([0; 1]). To show that j() s the Lebesgue measure, t s sucent by Proposton 2.7 to check that they have the same value on open sets. Snce any open set n [0; 1] s the countable unon of dsjont open (or half-open half-closed at 0 or 1) ntervals, t suces to check ths on such ntervals. But snce the dyadc numbers are dense n [0; 1], t s easy to see, for example, that ((x; y)) = sup q ((x; y)) = y x = ((x; y)): 3 The Normalsed Probablstc Power Doman In ths secton, we consder the subset of normalsed valuatons of the probablstc power doman and extend and sharpen the results n Subsecton 2.2 for ths subset. For any topologcal space Y, let P 1 Y P Y be the set of contnuous valuatons on Y whch are normalsed,.e. (Y ) = 1. Note that f Y has bottom?, then P 1 Y has bottom?. Let DCPO? denote the category of dcpo's wth bottom and contnuous maps. Dene P 1 on morphsms as for P,.e. for f : Y!, put P f() = f 1. Then P 1 : DCPO?! DCPO? s, lke P, a locally contnuous functor, whch we call the normalsed probablstc power doman functor. 10

11 We wll now extend the results of subsecton 2.2 to the normalsed power doman. In the rest of ths paper, we denote the way-below relatons n P Y and P 1 Y by and 1 respectvely. Let Y be a dcpo wth bottom. We get the analogue of Proposton 2.3. Proposton 3.1 For two smple valuatons 1 = b2b r b b 2 = c2c s c c n P 1 Y, we have: 1 v 2, for all b 2 B and all c 2 C, there exsts a nonnegatve number t b;c such that c2c and t b;c 6= 0 mples b v c. t b;c = r b b2b t b;c = s c Proof The `f' part follows from Proposton 2.3. For the `only f' part, we know from the `only f' part of that proposton that there exst nonnegatve numbers t b;c such that c2c t b;c = r b b2b t b;c s c and t b;c 6= 0 mples b v c. If for any c 2 C, we have P b t b;c < s c, then we obtan a contradcton, snce 1 = b r b = b It follows that P b t b;c = s c Dene the maps where and c t b;c = c for all c 2 C. b t b;c < c s c = 1: m + : P Y! P Y m : P Y! P Y 7! + 7! + (O) = ( (O) f O 6= Y 1 otherwse ( (O) f O 6= Y (O) = (Y f?g) otherwse. Note that Y f?g s an open set, and hence s well-dened. The map m + puts the mssng mass of on the bottom? of Y to produce a normalsed valuaton +. The map m removes any mass that may exst on?. The followng propertes are easy consequences of the dentons; the proofs are omtted. 11

12 Proposton 3.2 () The maps m + and m are well-dened, contnuous and satsfy: m + m + = m + w 1 m m = m v 1 m m + = m m + m = m + where 1 s the dentty map on P Y. () & 2 P 1 Y ) + 1. () 1 ) v. Corollary 3.3 If Y s an (!)-contnuous dcpo wth bottom, then P 1 Y s also an (!)-contnuous dcpo wth a bass of normalsed smple valuatons. Proof Note that P 1 Y s the mage of the contnuous dempotent functon m + : P Y! P Y. But the mage of any contnuous dempotent functon (.e. retract) on an (!)-contnuous dcpo s another (!)-contnuous dcpo [1, Theorem 3.14]. To prove an analogue of Proposton 2.5 for P 1 Y, we need a techncal lemma. Assume n the rest of ths secton that Y s a contnuous dcpo wth bottom. Lemma 3.4 Suppose ; 2 P 1 Y. Then 1 mples (Y f?g) < 1. Proof Assume 1. For n 1, let n = 1 n? + (1 1 ). We have n n (Y ) = 1 and n (O) = (1 1 )(O) for any n 1 and any open set O 6= Y. n It follows that h n n1 s an ncreasng chan wth lub. Therefore, v n for some n 1. We conclude that as requred. (Y f?g) n (Y f?g) = (1 1 n )(Y f?g) 1 1 n < 1 Proposton 3.5 For two smple valuatons 1 = b2b r b b 2 = c2c s c c n P 1 Y, we have 1 1 2? 2 B wth r? 6= 0, and, for all b 2 B and all c 2 C, there exsts a nonnegatve number t b;c wth t?;c 6= 0 such that and t b;c 6= 0 mples b c. t b;c = r b c2c b2b t b;c = s c 12

13 Proof For the `f' part, note that the condtons above mply, by Proposton 2.5, that = 1 r b b s c c = 2 : b2b f?g c2c Now Proposton 3.2() mples 1 = ( 1 ) For the `only f' part, rst note that by Lemma 3.4 we must have? 2 B wth r? 6= 0. Therefore by Proposton 3.2() = 1 s c c = 2 : b2b f?g r b b c2c Hence, by Proposton 2.5, there exsts, for each b 2 B f?g and c 2 C, a nonnegatve t b;c wth r b = c2c t b;c (b 6=?) s c > t b;c b6=? such that t b;c 6= 0 ) b c. Put t?;c = s c t b;c : b6=? Then, t s easy to check that c2c t?;c = c2c Furthermore we clearly have as requred. (s c t b;c ) = 1 b6=? b2b b6=?;c2c t b;c = t?;c + t b;c = s c b6=? 4 The Generalsed Remann Integral t b;c = 1 b6=? r b = r? : We wll now use the results of the prevous sectons to dene the generalsed Remann ntegral. In ths secton and n the rest of the paper, let f :! R be a bounded real-valued functon on a compact metrc space (; d) and let be a bounded Borel measure on. Let m = nf f[] and M = sup f[]. Wthout loss of generalty,.e. by a rescalng, we can assume that s normalsed. By Theorem 2.15, corresponds to a unque valuaton e() = s 1 2 S 1 () P 1 U, whch s supported n s[] and s, by Proposton 2.14, a maxmal element of P 1 U. Recall that we wrte nstead of e(). 13

14 4.1 The Lower and Upper R-Integrals We wll dene the generalsed Remann ntegral by usng generalsed Darboux sums as follows. Denton 4.1 For any smple valuaton = P b2b r b b 2 P U, the lower sum of f wth respect to s S` (f; ) = r b nf f[b]: b2b Smlarly, the upper sum of f wth respect to s S u (f; ) = b2b r b sup f[b]: Note that, snce f s bounded, the lower sum and the upper sum are well-dened real numbers. When t s clear from the context, we drop the subscrpt and smply wrte S`(f; ) and S u (f; ). Clearly, we always have S`(f; ) S u (f; ). Proposton 4.2 Let 1 ; 2 2 P 1 U be smple valuatons wth 1 v 2, then S`(f; 1 ) v S`(f; 2 ) and S u (f; 2 ) v S u (f; 1 ): Proof Assume 1 = b2b r b b and 2 = c2c s c c : Let t b;c be the nonnegatve numbers gven by Proposton 3.1. Then, S`(f; 1 ) = P b r bnf f[b] = P P b c t b;cnf f[b] P P b c t b;c nf f[c] = P P c b t b;c nf f[c] = P c s c nf f[c] = S`(f; 2 ): S u (f; 1 ) = P b r b sup f[b] = P P b c t b;c sup f[b] P P b c t b;csup f[c] = P P c b t b;csup f[c] = P c s csup f[c] = S u (f; 2 ): Note that, n the above proof, t s essental that the smple valuatons are normalsed,.e. that we work n P 1 U, to deduce that the upper sum decreases. The latter would not hold n general for smple valuatons n P U. Corollary 4.3 If 1 ; 2 2 P 1 U are smple valuatons wth 1 ; 2 1, then S`(f; 1 ) S u (f; 2 ). 14

15 Proof Snce the set of normalsed smple valuatons way-below n P 1 U s drected, there exsts a normalsed smple valuaton 3 2 P 1 U such that 1 ; 2 v 3 1. By Proposton 4.2, we therefore have S`(f; 1 ) S`(f; 3 ) S u (f; 3 ) S u (f; 2 ): Therefore, f we consder the drected set of smple valuatons way-below n P 1 U, then every lower sum s bounded by every upper sum. Ths s smlar to the stuaton whch arses for Darboux sums n Remann ntegraton. Denton 4.4 The lower R-ntegral of f wth respect to on s R fd = sup S` 1 (f; ): Smlarly, the upper R-ntegral of f wth respect to on s R Clearly, R R fd RR fd. fd = nf 1 S u (f; ): Denton 4.5 We say f s R-ntegrable wth respect to on, and wrte f 2 R () f ts lower and upper ntegrals concde. If f s R-ntegrable, then the R-ntegral of f s dened as R fd = R fd = R fd: When R there s no confuson, R we smply wrte R() nstead of R () and fd nstead of R fd (smlarly for the lower and upper R-ntegrals). The followng charactersaton of R-ntegrablty, smlar to the Lebesgue condton for the ordnary Remann ntegral, s an mmedate consequence of the denton. Proposton 4.6 (The R-condton.) We have f 2 R() for all > 0 there exsts a smple valuaton 2 P 1 U wth 1 such that S u (f; ) S`(f; ) < : There s also an equvalent charactersaton of the R-ntegral n terms of generalsed Remann sums wth ts well-known parallel n ordnary Remann ntegraton. Denton 4.7 For a smple valuaton = P b2b r b b 2 P U and for a choce of b 2 b for each b 2 B, the sum P b2b r bf( b ) s called a generalsed Remann sum for f wth respect to and s denoted by S (f; ). Note that we always have S`(f; ) S (f; ) S u (f; ) for any generalsed Remann sum S (f; ). We therefore mmedately obtan: 15

16 Proposton 4.8 We have f 2 R() wth R-ntegral value K for all > 0 there exsts a smple valuaton = b2b r b b 2 P 1 U wth 1 such that jk S (f; )j < for all generalsed Remann sums S (f; ) of f wth respect to. Havng dened the noton of R-ntegrablty wth respect to smple valuatons way below n P 1 U, we can now deduce the followng results. Proposton 4.9 If f s R-ntegrable and = F 0, where h 0!-chan n P U, then fd = lm S`(f; ) = lm S u (f; ) = lm S (f; );!1!1!1 s an where S (f; ) s any generalsed Remann sum of f wth respect. Proof Let > 0 be gven. Let the smple valuaton 2 P 1 U wth 1 be such that S u (f; ) S`(f; ) <. Snce = F +, there exsts N 0 such that v + and (U) > 1 for all N. Therefore, for all N, + = + r wth r = 1 (U) <. For N, we therefore have S`(f; + ) S`(f; ) = r nf f[] = r m and also the nequaltes S u (f; + ) Su (f; ) = r sup f[] = r M S`(f; ) S`(f; + ) Su (f; + ) Su (f; ) S`(f; ) fd S u (f; ): It follows that js`(f; + ) R fdj < and js u (f; + ) R fdj < for N. We conclude that for N, and the result follows. js`(f; ) js u (f; ) fdj < (1 + jmj) fdj < (1 + jmj); Corollary 4.10 If f s R-ntegrable and = F 0, where h 0 s an!-chan n P 1 U, then S`(f; ) ncreases to R fd and S u (f; ) decreases to R fd. 16

17 4.2 Elementary Propertes of the R-Integral We now show some smple propertes of the R-ntegral. Proposton 4.11 () If f; g 2 R() then f + g 2 R() and R (f + g) d = R fd + R gd. () If f 2 R() and c 2 R, then cf 2 R() and R cf d = c R fd. () More generally, f f; g 2 R() so s ther product h :! R wth h(x) = f(x)g(x). Proof We wll only prove (). For any nonempty compact subset b we have: sup (f + g)[b] sup f[b] + sup g[b] nf (f + g)[b] nf f[b] + nf g[b]: Hence, for any smple valuaton 1, we have S u (f + g; ) S u (f; ) + S u (g; ) S`(f + g; ) S`(f; ) + S`(g; ): Let > 0 be gven. There exst smple valuatons 1 ; 2 1 wth S u (f; 1 ) < fd + =2 S u (g; 2 ) < gd + =2: Let the smple valuaton be such that 1 ; 2 v 1. Then S u (f; ) S u (f; 1 ) and S u (g; ) S u (g; 2 ), and we have: S u (f + g; ) S u (f; ) + S u (g; ) S u (f; 1 ) + S u (g; 2 ) R fd + R gd + : Therefore, R (f +g) d R fd+ R gd. Smlarly, R fd+ R gd R (f +g) d, and the result follows. Gven the functon f :! R as before, dene two functons f + ; f :! R by f + (x) = ( f(x) f f(x) 0 0 otherwse and f (x) = ( f(x) f f(x) 0 0 otherwse. Proposton 4.12 If f 2 R(), then f + ; f 2 R() and R fd = R f + d R f d. Proof For any non-empty compact set b, we have and therefore sup f + [b] nf f + [b] sup f[b] nf f[b] S u (f + ; ) S`(f + ; ) S u (f; ) S`(f; ): By the R-condton (Proposton 4.6), f + 2 R(). Smlarly, f 2 R(). Snce f = f + f, by Proposton 4.11, we get R fd = R f + d R f d. 17

18 The followng propertes are easly shown. Proposton 4.13 () If f s nonnegatve and f 2 R() then R fd 0. () f 2 R() ) jfj 2 R() and j fdj jf jd: We wll make frequent use of the followng result n the next sectons. As before, let be a normalsed Borel measure on the compact metrc space. Proposton 4.14 Let h 2I be a drected set of smple valuatons = b2b r ;b b n P 1 U wth lub. Then for all > 0 and all > 0, there exsts 2 I wth r ;b < b2b ;jbj where jbj s the dameter of the compact set b. Proof Let > 0 and > 0 be gven. Let ha k k2j n, n 1, be the collecton of all open balls n wth radus at [ most 1=n, and put O n = a k : k2jn Then, for all n 1, we have s[] s[o n ] and hence (O n ) = (s[]) = 1, snce s supported n s[]. Now choose n > 2= so that 1=n < =2. Snce sup 2I (O n ) = (O n ) = 1, there exsts 2 I wth (O n ) > 1. It follows that r ;b < as requred. b2b ;jbj 5 R-ntegraton and Remann ntegraton In ths secton, we show that Remann ntegraton s equvalent to R-ntegraton on compact real ntervals wth respect to the Lebesgue measure. Let = [0; 1] be the unt real nterval wth the Lebesgue measure. Recall from Example 2.16 that any partton q : 0 = x 0 < x 1 < < x j 1 < x j < x j+1 < < x N 1 < x N = 1 of [0; 1] gves rse to a smple valuaton q = N j=1 r j bj where b j = [x j 1 ; x j ] and r j = x j x j 1. Gven any bounded functon f : [0; 1]! R, the lower and upper Darboux sums of f wth respect to 18

19 the partton q n the ordnary Remann ntegraton of f are precsely the lower and upper sums S`(f; ) and S u (f; ) of R-ntegraton. We wll of course use ths fact to show that f s Remann ntegrable f 2 R(), and that when the two ntegrals exst, then they are equal,.e. R [0;1] fd = 1 0 f(x)dx: However, t can be easly shown that q s not way-below f N > 1. In fact, for 1, let = 1 [0;1] + (1 1), where h 1 s the!-chan gven n Proposton Then, we have F 1 =, but there s no 1 wth q v, showng that q s not way-below. But, the followng lemma shows that t s possble to obtan a valuaton close to q whch s way-below. Lemma 5.1 Let q be any partton of [0; 1] nducng the smple valuaton q as above, and let 0 < < 1. Then, = [0;1] + (1 ) q 1. Proof By Proposton 3.2(), t suces to prove that (1 ) q. Choose a real number such that > > 0. For each j = 1; : : :; N, let b 0 = j [x j r 2 j; x j 1r 2 j]. Snce b 0 j (x j 1; x j ) b j, t follows that b j b 0 holds n U. Let j 0 = (1 ) P N r j=1 j b 0 j. By Proposton 2.5, wth t jj = (1 )r j and t jj 0 = 0 for j 6= j 0, we have (1 ) q 0. Snce the length of b 0 s (1 j )r j and the ntervals b 0 j, j = 1; : : :; N, are dsjont, t easly follows that 0 v. Thus, (1 ) q 0 v as requred. Theorem 5.2 A bounded real-valued functon on a compact real nterval s Remann ntegrable t s R-ntegrable wth respect to the Lebesgue measure. Furthermore, the two ntegrals are equal when they exst. Proof Assume wthout loss of generalty that the compact real nterval s n fact the unt nterval. For the `f' part, assume f : [0; 1]! R s R-ntegrable. Let > 0 be gven. By the R-condton, there exsts 1 wth S u (f; ) S`(f; ) <. Now take, for example, the!-chan h q 0 of smple valuatons of Proposton 2.17 whose lub s. Snce q 2 P 1 U[0; 1] for all 0, there s 0 wth v q. Hence S u (f; q ) S`(f; q ) S u (f; ) S`(f; ) < and therefore the Remann condton s satsed and f s Remann ntegrable. Snce, by Corollary 4.10, the R-ntegral s the supremum of S`(f; q ) and the latter s also the Remann ntegral of f, we conclude that the two ntegrals are equal when they exst. For the `only f' part, assume f s Remann ntegrable and let q : 0 = x 0 < x 1 < < x N = 1 be a partton of [0; 1] wth S u (f; q ) S`(f; q ) <. By Lemma 5.1, = [0;1] + (1 ) q s way-below. We also have S u (f; ) S`(f; ) (M m) + (1 ) < (M m + 1): We conclude that f satses the R-condton, and s therefore R-ntegrable. 19

20 We conclude that ordnary Remann ntegraton s a partcular nstance of R-ntegraton. 6 Further Propertes of R-ntegraton In ths secton, we wll show that all the basc results for ordnary Remann ntegraton on a compact nterval n R can be extended to R-ntegraton. Assume as before that s a normalsed Borel measure on the compact metrc space. We rst show that contnuous functons are R-ntegrable. Theorem 6.1 Any contnuous functon f :! R s R-ntegrable wth respect to. Proof Let > 0 be gven. By the unform contnuty of f on the compact set, there exsts > 0 such that d(x; y) < mples jf(x) f(y)j <. 2 By Proposton 4.14, there exsts a smple valuaton = P b2b r b b wth 1 such that r b 2(M m + 1) : Therefore, jbj S u (f; ) S`(f; ) = b2br b (supf[b] nff[b]) = b2b;jbj< < 2 + (M m) 2(M m+1) < : r b (supf[b] nff[b]) + It follows by the R-condton that f 2 R(). b2b;jbj r b (supf[b] nff[b]) Next we wll prove that a bounded functon s R-ntegrable wth respect to a Borel measure f and only f ts set of dscontnutes has measure zero. Ths wll generalse the well-known Lebesgue crteron for ordnary Remann ntegraton of a bounded functon on a compact real nterval. We need some dentons and propertes relatng to the oscllaton of a bounded functon f :! R whch generalse those of a bounded functon on a compact real nterval as presented, for example, n [2]. For convenence and consstency, we wll use the termnology n that work. Denton 6.2 Let T. The number f (T ) = supff(x) f(y) j x; y 2 T g s called the oscllaton of f on T. For x 2, the number! f (x) = lm h!0 + f(b(x; h)) 20

21 where B(x; h) s the open ball of radus h > 0 at x, s the oscllaton of f at x. For each r > 0, let D r = fx 2 j! f (x) 1=rg: The followng propertes then are straghtforward generalsatons of those n [2, pages ]. Proposton 6.3 () f s contnuous at x 2! f (x) = 0. () If! f (x) < for all x 2, then there exsts > 0 such that for all compact subsets b wth jbj < we have f (b) <. () For any r > 0 the set D r s closed. If D s the set of dscontnutes of f, then usng Denton 6.2 and Proposton 6.3(), we can wrte D = S n1 D n where D 1 D 2 D 3 : : : s an ncreasng chan of closed sets. Hence, D s an F, and therefore a Borel, set. Recall that we assume to be a normalsed measure on the compact metrc space. Lemma 6.4 Let d be compact, and let = P b2b r b b be a smple valuaton n P 1 U. () If v then b\d6=; r b (d): () If 1, then where b s the nteror of b. b \d6=; r b (d) Proof () We have P b\d6=; r b = 1 P b\d=; r b snce (U) = 1 = 1 P b d r b = 1 P b2( d) r b = 1 (( d)) 1 (( d)) snce v = 1 ((s[]) (s[d])) snce s supported n s[] = (d) () By the nterpolatve property of 1, there exsts a normalsed smple valuaton = P c2c s c c such that 1 1. Let t b;c be gven as n 21

22 Proposton 3.5. Then (d) P c\d6=; s c by part () = P P c\d6=; bc t b;c P c2c t b;c P b \d6=; = P b \d6=; r b: Theorem 6.5 A bounded real-valued functon on a compact metrc space s R-ntegrable wth respect to a bounded Borel measure ts set of dscontnutes has measure zero. Proof Necessty. Let D be the set of dscontnutes of f :! R and suppose (D) > 0. Snce D = S n1 D n, we must have (D n ) > 0 for some n 1. Fx such n and let = P b2b r b b be any smple valuaton wth 1. Then S u (f; ) S`(f; ) = b r b (supf[b] nff[b]) b \Dn6=; b \Dn6=; r b (supf[b] nff[b]) r b =n by denton of D n (D n )=n > 0: by Lemma 6.4() Therefore, f does not satsfy the R-condton and s not R-ntegrable. Sucency. Assume (D) = 0. It follows that (D n ) = 0 for all n 1. Fx n 1. Snce s regular, there exsts an open set v 2 wth D n v and (v) < 1=n. Choose an open set w 2 whch contans D n and whose closure s contaned n v. Let 1 > 0 be the mnmum dstance between v and the closure of w. For x 2 w we have! f (x) < 1=n. Therefore, by Proposton 6.3() appled to w, there exsts 2 > 0 such that for any compact subset c w wth jcj 2 we have f (c) < 1=n: (1) Let 0 < < mn( 1 ; 2 ). By Proposton 4.14, there exsts a smple valuaton = P b2b r b b wth 1 such that jbj r b < 1=n: (2) Observe that f jbj <, then b s contaned n at least one of the sets v or w. We also have (v) (v) = (v) < 1=n (3) 22

23 snce s supported n s[]. Therefore, S u (f; ) S`(f; ) = b2br b (supf[b] nff[b]) jbj M m n + jbj;bv + M m n 2(M m) + 1 : n + + jbj;b w Snce n 1 s arbtrary, f 2 R() by the R-condton. jbj;b w r b n by (2), (3) and (1) If s a bounded Borel measure on and C s a closed subset, then the restrcton C s R a bounded Borel measure on C. For convenence, we wrte f 2 R C () and C fd, f f s R-ntegrable on C wth respect to ths restrcton. Corollary 6.6 If f 2 R () then f 2 R C () for all closed subsets C. Proposton 6.7 If f 2 R () and C; D are closed subsets wth C \ D = ;, then C[D fd = C fd + D fd: Proof By Corollary 6.6, we know that the three ntegrals exst. Let h 0 be an!-chan of smple valuatons n P UC wth lub C. Smlarly, let h 0 be an!-chan of smple valuatons n P UD wth lub D. Then, t s straghtforward to check that h + 0 s an!-chan of smple valuatons n P U(C [ D) wth lub C[D. For each 0, we have Therefore, S` C[D (f; + ) = S` C (f; ) + S` D (f; ): RC[D fd = lm!1 S` C[D (f; + ) = lm!1 S` C (f; ) + S` D (f; ) = lm!1 S` C (f; ) + lm!1 S` D (f; ) = R C fd + R D fd: Next, we consder the R-ntegrablty of the unform lmt of a sequence of R-ntegrable functons. Theorem 6.8 If the sequence hf n n0 of R-ntegrable functons f n :! R s unformly convergent to f :! R, then f s R-ntegrable and R fd = lm!1 R f d. 23

24 Proof Let > 0 be gven. We show that f satses the R-condton. Let N 0 be such that jf n (x) f(x)j < =3 for all n N and all x 2. Then for all smple valuatons 2 P 1 U we have js u (f f N ; )j < =3 and js`(f f N ; )j < =3. Snce f N s R-ntegrable, there exsts a smple valuaton 1 wth S u (f N ; ) S`(f N ; ) < =3. Therefore, S u (f; ) S`(f; ) S u (f f N ; ) + S u (f N ; ) S`(f f N ; ) S`(f N ; ) js u (f f N ; )j + js`(f f N ; )j + S u (f N ; ) S`(f N ; ) < = and, hence, f 2 R(). Furthermore, for all n N, we have j R fd R f n dj = j R f f n dj R jf f n jd : In the next secton, we wll also show the generalsaton of Arzela's theorem for R-ntegraton. 7 R-Integraton and Lebesgue Integraton It s well known that when the ordnary Remann ntegral of a bounded functon on a compact real nterval exsts, so does ts Lebesgue ntegral and the two ntegrals concde. In ths secton, we wll show that ths result extends to R-ntegraton. In order to show that an R-ntegrable functon s Lebesgue ntegrable, we construct an ncreasng sequence of smple measurable functons whch tend to our functon. We do ths by consderng the set of deatons on U. Recall [18] that a deaton on a dcpo Y s a contnuous map d : Y! Y whch s below the dentty d v 1 Y and ts mage m(d) s nte. If b 2 B = m(d), then D = d 1 (b) satses the followng propertes: () x v y v z & x; z 2 D ) y 2 D. () For any drected set hx 2I 2 I. wth F x 2 D, we have x 2 D for some () For any drected set hx 2I wth x 2 D for all 2 I, we have F x 2 D. 24

25 It follows [24] that D s a crescent,.e. D = v w for some open sets v; w 2 Y. Now consder the map P d : P Y! P Y, nduced by the probablstc power doman functor P on the deaton d v 1 Y. We have P d() = d 1 v, snce d 1 (O) O for all O 2 Y. Consder the unque extenson of 2 P Y to the rng generated by the open sets,.e. put (D) = (v) (v \ w) for each crescent D = v w. Then t s easly seen that d 1 = r b b b2b wth r b = (d 1 (b)). Hence, for each deaton d and each contnuous valuaton on Y, we obtan a smple valuaton d 1 below. Note that f s normalsed so s d 1. Recall also that f Y s the retract of an SFP doman, then the set of deatons way-below the dentty map s drected and has the dentty as ts lub [18, page 88]. We can now deduce: Proposton 7.1 Any contnuous valuaton on a retract of an SFP doman s the lub of an!-chan of smple valuatons nduced from deatons below the dentty map. Proof Let Y be a retract of an SFP doman and be a contnuous valuaton on Y. By the above remark, there exsts an!-chan hd 0 of deatons on Y wth 1 Y = F d. By the local contnuty of P (Proposton 2.2), we have 1 P Y = F P d and therefore = G P d () = G d 1 as requred. We are now n a poston to prove the man result n ths secton. For clarty we denote the Lebesgue ntegral of a real-valued functon f :! R wth R R respect to the Borel measure by L fd and the R-ntegral by R fd. We also drop the subscrpt. Theorem 7.2 If a bounded real-valued functon f s R-ntegrable wth respect to a Borel measure on a compact metrc space, then t s also Lebesgue ntegrable and the two ntegrals concde. Proof Snce U s an!-contnuous bounded complete dcpo wth bottom, there exsts by Proposton 7.1 an!-chan hd 0 of deatons d : U! U wth F d 1 = and each 0 nduces a smple valuaton = d 1 = r ;b b b2b where B = md and r ;b = (d 1 (b)). For each 0, dene two functons f :! R f + :! R x 7! nff[s 1 (d 1 (d (s(x))))] x 7! supf[s 1 (d 1 (d (s(x))))] where s :! U s, as before, the sngleton map. Snce for each 0 and x 2, we have d 1 (d (s(x))) = v w for some open sets v; w 2 U, 25

26 t follows easly that s 1 (d 1 (d (s(x)))) = s 1 (v) s 1 (w) s a crescent of. Moreover, as the mage of d s nte, s parttoned to a nte number of such crescents. Therefore, f and f + are smple measurable functons. Is s easy to see that for each x 2 we have: m : : : f (x) f +1 (x) : : : f(x) : : : f + +1 (x) f + (x) : : : M where m and M are, as before, the nmum and the supremum of f on. Let f :! R f + :! R x 7! lm!1 f (x) x 7! lm!1 f + (x): Then f (x) f(x) f + (x) for all x 2. By the monotone convergence theorem, f and f are Lebesgue ntegrable. We wll calculate ther Lebesgue ntegrals. For each b 2 B, let ;b = supf[s 1 (d 1 (b))] ;b = nff[s 1 (d 1 (b))]: Snce d 1 (b) "b, we have s 1 (d 1 (b)) b. Hence, for all 0 and b 2 B, nff[b] ;b ;b supf[b]: We can now obtan the followng estmates for the Lebesgue ntegrals of f and f + : L R f + d = P b2b r ;b ;b P b2b r ;b supf[b] and = S u (f; ) L R f d = P b2b r ;b ;b P b2b r ;b nff[b] = S`(f; ): R R Snce f f + mples L f d L f + S`(f; ) L f d, we obtan: d L f + d Su (f; ): As f s assumed to be R-ntegrable, we know by Propostons 4.2 and 4.9 that S`(f; ) ncreases to R R fd and S u (f; ) decreases to R R fd. Therefore, L f d! R fd and L f + d! R as! 1. By the monotone convergence theorem, we have: 26 fd

27 L L f d = lm!1 L f + d = lm!1 L f f + d = R d = R fd fd: It now follows that R L (f + f ) d = 0 whch mples that f + = f almost everywhere. Therefore f = f = f + almost everywhere. We conclude that f s Lebesgue ntegrable and L fd = L f d = L f + d = R fd as requred. We can now also obtan the generalsaton of Arzela's theorem for R-ntegraton. Corollary 7.3 Suppose the sequence hf n n0 of real-valued and unformly bounded functons on s pontwse convergent to an R ntegrable functon f. Then, we have: R fd = lm n!1 R f n d: Proof Ths follows mmedately by applyng Theorem 7.2 and usng the Lebesgue domnated convergence theorem. 8 Applcatons to Fractals An mmedate area of applcaton for R-ntegraton s n the theory of terated functon systems (IFS) wth probabltes. Recall [15, 3] that an IFS wth probabltes, f; f 1 ; : : :; f N ; p 1 ; : : :; p N g, s gven by a nte number of contractng maps f :! (1 N) on a compact metrc space, such that each f s assgned a probablty weght p wth 0 < p < 1 and N =1 p = 1: An IFS wth probabltes gves rse to a unque nvarant Borel measure on. If R n, then the support of ths measure s usually a fractal.e. t has ne, complcated and non-smooth local structure, some form of self-smlarty and, usually, a non-ntegral Hausdor dmenson. Conversely, gven any mage regarded as a compact set n the plane, one uses a self-tlng of the mage and Barnsley's collage theorem to nd an IFS wth contractng ane transformatons, whose attractor approxmates the mage. The theory has many applcatons ncludng n statstcal physcs [14, 6, 10], neural nets [5, 8] and mage compresson [3, 4]. 27

28 It was shown n [9, Theorem 6.2], that the unque nvarant measure of an IFS wth probabltes as above s the xed pont of the map T : P 1 U! P 1 U 7! T () dened by T ()(O) = P N =1 p (f 1 (O)). Ths xed pont can be wrtten as F m0 m where 0 = and for m 1, m = T m ( ) = N 1 ; 2 ;:::;m=1 p 1 p 2 : : : p m f1 f 2 :::f m (): Therefore, the unque nvarant measure of the IFS wth probabltes s the lub of an!-chan of smple valuatons n P 1 U. Ths provdes a better algorthm for fractal mage decompresson usng measures [11], compared to the algorthms presented n [4]. Suppose now we have a bounded functon f :! R whose set of dscontnutes has -measure zero, then we know that ts Lebesgue ntegral wth respect to concdes wth ts R-ntegral wth respect to. Fx x 2 and, for each m 1, consder the generalsed Remann sum S x (f; m ) = N 1 ; 2 ;:::;m=1 From Proposton 4.9, we mmedately obtan: p 1 p 2 : : : p mf(f 1 f 2 : : :f m(x)): Theorem 8.1 For an IFS wth probabltes and a bounded real-valued functon f whch s contnuous almost everywhere wth respect to the nvarant measure of the IFS, we have for any x 2. L f d = R f d = lm m!1 S x(f; m ); If f satses a Lpschtz condton, R then, for any > 0, we can obtan a nte algorthm to estmate f d up to accuracy [11]. The only other method for computng the ntegral s by Elton's ergodc theorem [12]: The tme-average of f wth respect to the non-determnstc dynamcal system f 1 ; f 2 ; : : :; f N :!, where at each stage n the orbt of a pont the map f s selected wth probablty p, tends, wth probablty one, to ts space-average,.e. to ts ntegral. However, n ths case, the convergence s only wth probablty one and there s no estmate for the rate of convergence. Therefore, the above theorem provdes a better way of computng the ntegral. 28

29 Example 8.2 Fnally, we consder a concrete example. Let C = f1; 2; ; Ng! be the Cantor space wth the followng metrc d(x; y) = 1 n=0 (x n ; y n ) 2 n where the Kronecker delta s gven by ( 0 f k = l (k; l) = 1 otherwse. Ths metrc s equvalent to the Cantor (product) topology, and s frequently used n mathematcs and theoretcal physcs. Let fc; f 1 ; : : :; f N ; p 1 ; : : :; p N g be an IFS wth probabltes on C, wth f k : C! C x 7! kx; where kx s concatenaton of k and x. Its unque nvarant measure s dened on the closed-open subset by In fact, we have Let [ 1 2 m ] = fx 2 C j x j = j ; 1 j mg m = T m ( ) = ([ 1 2 m ]) = p 1 p 2 p m: N 1 ; 2 ;:::;m=1 f : C! R x 7! d(x; 1! ) p 1 p 2 : : : p m [1 2 :::m]: be the functon whch gves the dstance of the pont x to the pont 1!. Ths functon s contnuous and therefore ts Lebesgue ntegral wth respect to concdes wth ts R-ntegral wth respect to. The ntegral n fact represents the average dstance n C from 1! wth respect to the nvarant measure. The R-ntegral s easly obtaned usng Theorem 8.1 above wth x = 1!. In fact a straghtforward calculaton shows that S 1!(f; m ) = 2(1 1 2 m )(1 p 1)! 2(1 p 1 ) as m! 1. Therefore, L R fd = R R fd = 2(1 p 1 ). Acknowledgements I would lke to thank Achm Jung, Klaus Kemel and Phlpp Sunderhauf for dscussons on contnuous domans. I am n partcular grateful to Achm Jung who suggested the dea of usng deatons leadng to Proposton 7.1. Many thanks to Renhold Heckmann for suggestng shorter proofs for two lemmas. Ths work has been funded by the SERC \Foundatonal Structures for Computer Scence" at Imperal College. 29

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