Neubrunn, Tibor

Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaTibor NeubrunnAlmost uniform convergence for continuous parametersMathematica Slovaca, Vol. 28 (1978), No. 3, 321--328Persistent URL: http://dml.cz/dmlcz/132374Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1978Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMath. Slovaca 28,1978, No. 3, 321—328 ALMOST UNIFORM CONVERGENCE FOR CONTINUOUS PARAMETERS TIBOR NEUBRUNN Various papers deal with Jegoroff's theorem of almost uniform convergence. The classical version proved in [3] (see e.g. [2] p. 88) concerns sequences of functions. It is well known that it fails if instead of sequences a system {/'} (t e T) of functions is considered. More precisely, a function /(*, t) on X x T is given and by means of this function the collection {/'} (t e T) where / ' are t — sections of t is considered. The counterexamples were given in [11], [12] (see also [10]). If the notion of the almost uniform convergence is weakened, then a weaker analogy of Jegoroff's theorem may be obtained also for continuous parameters. Such results were proved in [13] and [14]. On the other hand, there is a possibility to obtain also for a continuous parameter the classical version of Jegoroff's theorem, if certain assumption on /(*, t) as a function of two variables are given. Thus, e.g., the Borel measurability of / as a function of two real variables is sufficient, as was proved by Tolstoff [11], using the properties of the analytic sets in the plane. In the present paper we give two theorems of this kind covering the cases when X and T are sufficiently general spaces. Of course the proofs will differ from those of Tolstoff. Definition 1. Let (X, 5 ,̂ JU) be a measure space (in the sense of [2]) and T a topological space. Let F: Xx T—»JR be a function and t0eTa point. A system {/'} (t 6 T), where / '(*) = /(.*, t) for xeXis said to be almost uniformly conver-gent to a function <p defined on X, if to any e>Q there exists a set E effsuch that li(E)<e and the system {/'} converges uniformly to <p on X - East tends to t0. Our first result concerning the almost uniform convergence uses the notion of the quasicontinuity. Definition 2. If X, Y are topological spaces and g:X-^>Y a mapping, then g is said to be quasicontinuous x{) e X if for every two open sets U, V such that x{) e U, 321 g(x0)e V, there exists G open, G^=0, GaU and g(G)aV. It is said to be quasicontinuous on X if it is quasicontinuous at any x0eX.. Remark 1. The notion of quasicontinuity was used already in [4]. The relations between quasicontinuity continuity and other types of continuities was discussed in [6], [7], [8] and elsewhere. The characterization of discontinuity points of a quasicontinuous function was given in [5]. It seems to be worth mentioning that there exists a quasicontinuous function which is not Lebesgue measurable (see [6]). Lemma 1. If g: X—> Yis a quasicontinuous function on X, then the following is true :IfZaX is any open set and DczZis dense in Z, then f(D) is dense in f(Z). Proof. Let yef(Z) and xeZ such that f(x) = y. Let V be any open set containing y. We have from the quasicontinuity at x that a nonempty open GaZ exists with f(G) a V. Since GnD + 0 and f(Gr\D) c V, a point v eV belonging to f(D) exists. The proof is finished. Remark 2. It is known that the converse of Lemma 1 holds too. However, we do not use this fact. In what follows, we suppose that the functions / which are dealt with assume real values. A generalization for metric spaces or some uniform spaces is possible. But we are of the opinion that in that direction a sufficiently general version is given in [14]. By the same method as in [14] our results may be transferred to suitable uniform spaces. As to the spaces X, T on the product X x T of which the function / will be defined, we shall suppose that (X, Sf, p) is a totally finite measure space and T a separable topological space satisfying the first countability axiom. These conditions will not be repeated in the formulations of the theorems. Only additional conditions if necessary, will be explicitely stated. It can be easily verified that the first countability axiom may be weakened in some of the results. The same is true for the total finiteness of the space (X, Sf9 (i) if we restrict ourselves to functions defined on E x T, where Ee^f, and y(E)< <». ' Lemma 2. Let f: X x T-+R. Let t0 e T. If for ever teT the functions / ' are measurable and lim f(x) = cp(x) for every x e X, then cp is a measurable function. t-*t0 Proof. From the assumption of the first countability of T it follows that q)(x) = = lim/(jc, tn). Hence q> is measurable as a limit of a sequence of measurable n—>oo functions. Remark 3. Obviously the assumption lim/'(jt) = cp(jc) for every xeX may be substituded by the assumption that lim/'(x) = q>(x) almost everywhere. t-*t0 322 Lemma 3. Let / : X x T-> R be such thatf are measurable for every t e Tand fx quasicontinuous for every x eX. Suppose that lim f(x) = <p(jc) for every xeX. t-*t0 Then for every open set GczTand any a >0 the set F—{x: |/(JC, t) — <P(JC)| ^ e , for every teG} is measurable. Proof. Let D be a coutable dense subset of G. Suppose that D is the set of values of a sequence {t*}r=i. Put Fl = {x:\f(x,tk)-q>(x)\^e9 for k = l, 2, . . .} . Since /'* are measurable functions for k = 1, 2, ... and <p is measurable according to Lemma 2, we have that Fu as a countable intersection of measurable sets, is measurable. It is sufficient to show that F = Fi. The inclusion F c F i is obvious. Let x eFy and t any point in G. According to Lemma 1, we have that fx(D) is dense in fx(G). Hence a sequence {tkn}n=l exists such that fx(tkn) tends to /(JC, t). Since JC eF1? it follows |/(JC, tkn)-q)(x)\^e for n = 1, 2, ... Hence |/(JC, t) - <p(jc)|-=i£. Since t was arbitrary, we have JC eF . The inclusion Ftc=F is proved. Theorem 1. Let f:Xx Y-+R. Let t0eTand lim f(x, t) = <p(jc) almost eve-t-*t0 rywhere on X. Let fx be quasicontinuous for every xeX. Then for t —> t0, f tends to <p almost uniformly. Proof. Without loss of generality we may suppose that f(x) converges to <p(jc) for every x eX. Let V- => V2 =>... be a base of open neighbourhoods at the point t0. Le t£>0. Put fork = 1,2, ...,n = 1,2, ...Enk = {JC:JCEX, |/(JC, t) - <p(jc)| -S - , n for every teV*}. According to Lemma 3 the sets Enk are measurable. Since oo lim/r(jc) = <p(jc) for every jceX, we have [jEk=X for n = l, 2, . . . Moreover t-*tn k = i EkczEnk+l for k = l ,2 , ... Thus for every n a number k(n) exists such that li(X-Enk(n))<^. Put E = H Enk(n). The fact that ii(X-E)<e follows now in the same way as in the proof of the classical Jegoroff's theorem, as well as the fact that the convergence is uniform on E is now quite similar to the one in the classical case. If r/ > 0 is arbitrary, then we can choose n such that — <rj. Now, since EczEnk{nh we have |/(*, t) - <p(*)| _S-for every t e Vk(n) and every xeE. 323 In this section another sufficient condition will be given for an almost uniform convergence in the case of a continuous parameter. The proof will be based on a method which was used for proving the measurability of functions of two variables by the author in his thesis (1963) and then applied in [1]. At first we shall give some notes concerning this method. The functions will be again real valued. The assumption concerning the domain will be specified in each of the assertions which follow. Definition 3. / / X is a nonempty set, then a collection & = {Pk} of nonempty sets k = 1, 2, ... n e N(k), where N(k) is either a set {1, 2, ..., nk} or the set of all positive integers, and [JPk =X for k = l,2, ..., is said to be a SP-system on X. n Remark 5. Note that in [1] we used.the notion of a ^-system only in the case when a measurable space (X, 9*) was considered and we supposed SPk e &*. This will not be the case here in general. U2Pke£f for n e N(k) and k = 1, 2, ..., we say that ^ is a measurable ^-system. Definition 4. If 3> is a SP-system on X, then a function f defined in X is said to be regular atx0 with respect to 2P, provided that for any open G containing f(x0) there exists k0 such that if k > k0, then x,x0ePk for some n implies f(x) eG. It is said to be regular on X with respect to $P if it is regular a t any x0 e X, with respect to $P. Definition 5. If X is a topological space with the topology ST and SP a 3P-system on X, then & is said to be regular with respect to ST provided that for any U e ST and any xeU there exists k0 such that if k > k0, then xe$Pk implies 2Pk <= U. Lemma 4. if <3P is a SP-system on a set X, then a function fonX which is regular on Xrelative to SP, is measurable with respect to the o-algebra generated by<3>. Proof. Let G be an open set. If x0ef~l(G), then there exists k0 such that if k>k0 and x, x0eSPn, then f(x) e G. Since for x0 there is n such that x0 e $Pn, we have for this n f(SPk)G if k > k0. Thus to any x0ef~l(G) a set $Pk may be associated such that f(8Pk) <= G. This means that f~\G) is a union of some sets belonging to £P. Corollary. If X is a topological space and SP a SP-system on X which is Borel measurable (i.e. every element of SP belongs to the o-algebra generated by all open sets), then any function regular on X with respect to $P is Borel measurable. Lemma 5. J/ X is a topological space with the topology ST and $P a regular SP-system on X, then the o-algebra generated by & contains the collection of all Borel sets. > Proof. It is sufficient to prove that any open set U belongs to the a-algebra 324 generated by 0*. Let xeU. From the regularity of 9 it follows that there exists k0 such that if k > k0 and x e &n> then 3>k c U. Since to any x and any k there exists n such that JC e £Pk. we have that U is a (countable) union of sets belonging to 3P. The lemma is proved. Corollary. If X is a topological space and 9 a regular, Borel measurable ^-system on X, then the o-algebra generated by 0* concides with the o-algebra of all Borel sets. Lemma 6. If X is a topological space and 9 a regular ̂ -system on X, then any continous function on X is regular on X with respect to 9. We omit the simple proof. It is contained in [1]. Lemma 7. Let Xbe a set and 9 a ^-system on X. Let fbea function regular on X with respect to 0>. Then if 0 =£ Y c X , there exists a SP-system 9* on Ysuch that the restriction f\Y is regular on Y with respect to 9*. Proof. If 0> = {0*}, A: = 1,2,...; neN(k), then put Qk = &knY for k = = 1,2,...; n eN(k). We may suppose that the sets Qk are nonempty. If this is not the case, the only thing which will be different is that N(k) will be sub-stituted by another finite set which is a subset of N(k). Evidently &* = {Qkn} k = 1, 2, ...; n eN(k) is a ^-system on Y. Now \i x0eY and G is an open set containing (f\ Y) (JC0), we have f(x0)eG. Since / is regular with respect to 2P, there is k0 such that if k>k0x, j c 0 e ^ , then/(x)eG. Since Q*cz£p*, we obtain inmediately that if k>k0 and JC, Jt0eQ*, then (f\ Y) (x)eG. Remark 6. It is evident from the proof of Lemma 7 that if a class 9 of functions and a ^-system 9 are given such that each fe £F is regular on X with respect to 9, then the class &>* of all /1 Y, / e & is regular on Y with respect to the £P-system 9,s constructed in the proof of Lemma 7. In what follows let (X, 5 ,̂ \i) be again a totally finite measure space, T a topological space and t0 e T a point possessing a countable base of neighbourhoods. Theorem 2. Let f:XxT->R. Let a ^-system & on T exist such that fx is regular on T with respect to the SP-system for every xeXand let f be measurable for every teT. Then if lim /(JC, t) = w(x) for almost every x, the convergence is t-*t0 almost uniform as t tends to t0. Proof. We may again suppose that lim/(jc, t) = q)(x) for every JCGX. Let r—>r0 { Vk}k^i be a decreasing sequence of open sets forming a base at t0. Exactly as in Theorem 1 the main thing is to prove that the sets El = {x: |/(x, t) - <p(jc)| ^ —, 325 for every teVk) are measurable for n = 1, 2, ..., k = 1, 2, ... Thus it is sufficient to prove that the set E = {x: \f(x, t) - y(x)\ ^ e , for every t e V} is measurable for any e >0 , and any open set Vc= T. Construct now a ^-system ^ * o n V such that for the elements of & we take the sets OJ = ^ n V . . ^ e ^ (see Lemma 7 and Remark 6). Choose in every nonempty Qk a point tk and denote by D the set of all these points. Since D is countable, the set F, where F={x: f(x, t) — q)(x)\^e, for every t eD} , is measurable as a countable intersection of measurable sets. (The measurability of each of the sets {x : |f(x, t) — qp(x)| § e} for a fixed t follows from the assumption and from the measurability of cp, which in its turn follows in the same way as in Lemma 2). Now we prove E=F. The inclusion E c F is trivial. Let xeF and te V be any point. Let {ty}r=i be a decreasing sequence of positive numbers converging to 0. Since fx is regular (on V) with respect to 0>*, there exists for any r/, a positive integer k(i) such that if k>k(i), then for any tx such that tu teQ* we have |/*(ti) - /*(t)|<ty. Especially if we choose instead of tx tk(i)eD, depending on /(t*(0 = t(i), to simplify^the notation), which belong to the same Qk as t, we have \fx(tk(i)) - /*(t)| <ty. Thus a sequence of points tk(i) = t(i) belonging to D exists, such that lim/x(f(i)) = fx(t). Since t(i)eD, / = 1, 2, ..., we have \f(x,t(i))-(p(x)\^e fori = 1,2, .... hence \f(x, t) - q>(x)\ ;5 e. Since for t e V we may choose any point from V we have |/(JC, t) — q>(x)\ .§ e for any t e V, and s o x e E . Hence E — F and E is measurable. Thus the measurability of Enk, k = 1, 2,., n eN(k), is proved and the rest of the proof proceeds as in Theorem 1. Corollary. Let (X, 5 ,̂ p) be a totally finite measure space and T a second countable topological space. Let f: X x T-*R be such that fx are continuous for every xeX and f measurable for every teT. Then if lim ff (x) = w(x) for almost every x e X, the convergence is almost uniform as t tends to t0. Proof. If T is a second countable topological space, then a regular ^-system on T exists (See [1]). Now according to Lemma 6 the sections fx are regular with respect to ^ and the result follows. This section will be devoted to some discussion and lpossible generalization of the obtained results. The aim of Theorem 1 and Theorem 2 was to state some simple sufficient condition for the validity of Jegoroff 's theorem. It is not difficult to give an abstract formulation of the mentioned theorems. 326 Definition 6. We shall say that a function f defined on XxT (T a topological space) satisfies the property (S) if the following is true: (S) There exists a countable dense setDaT such that for any open GaT and any closed interval I {x:f(x,t)el, forall teG} = {x:f(x,t)eI, forallteGnD} Remark 7. Note that the property (S) follows from the assumptions of Theorem 1 (See Lemma 3) as well as from the assumptions of Theorem 2. Lemma 8. Iffis a real function defined onXx Tand statisfying (S) and cp a real function on X, then there exists a countable dense set D cz Tsuch that for any real ciSO and any open set G (SO holds: (SO {x: \f(x,t)-cp(x)\^c} forall teG = {x:\f(x,t)-cp(x)\^c forall teGnD} . Using the property (SO we can prove a lemma analogous to Lemma 3 and then the proof of the following theorem is strainghtforward. Theorem 3. Let (X, S),[i)be totally finite measure space. Let Tbea separable topological space satisfying the first countability axiom. Let the property (SO be satisfied. Then, if lim f(x, t) = q>(x) for almost every xeX, the convergence is t-*t0 almost uniform as t tends to t0. In the case in which T is a subspace of ( — oo, oo) with the usual topology and (X, &,(*)& probability space, the property (S) is the usual definition of a separable stochastic process. Hence from Theorem 3 and Lemma 8 we obtain as a corollary the following result. Corollary. If (X, &>, /x) is a probability space Tcz (- oo, oo) and f:Xx T^R a separable stochastic process, then the almost everywhere convergence off(x, t) to (p(x) for t—>t0 implies the almost uniform convergence. Remark 8. In paper [9] Jegoroffs theorem for sequences was proved in the case when the measure space was substituted by a space with the system of collections of „small measure sets". It is easy to see that our theorems may be proved also for such spaces. 327 REFERENCES • [1] DRAVECKÝ, J.—NEUBRUNN, T.: Measurability of functions of two variables. Mat. Čas., 23, 1973, 147—157. [2] HALMOS, P, R.: Measuгe Theory. Princeton 1968. [3] EGOROFF, D. F.: Suг les suites des fonctions mesuгables. C R. Acad. Sci Paris, 152, 1911. [4] KEMPISTY, S.: Sur les fonctions quasicontinues. Fund. Math., 19, 1932, 184—197. [5] LIPIŃSKI, J. S.—ŠALÁT, T.: On the points of quasicontinuity and cliquishness of functions. Czechosl. Math. J., 21 (96), 1971, 484—490. [6] MARCUS, S.: Suг les fonctions quasicontinues au sens de S. Kempisty. Coll. Math., 8, 1961, 47—53. [7] NEUBRUNNOVÁ, A.: On quasicontinuous and cliquish functions. Čas. Pěst. Mat., 99, 1974, 109—114. [8] NEUBRUNNOVÀ, A.: On ceгtain geneгalizations of the notion of continuity. Mat. Čas., 23, 1973, 374—380. [9] RIEČAN, B.: On abstгact foгmulation of some theorems of measuгe theory. Mat. fyz. Čas., 16, 1966, 268—273. [10] ŠTERBOVÁ, D.: A note on the generalization of Jegoroffs theorem. Mat. Čas., 18, 1968, 83—89. [11] TOЛCTOB, Г.: Зaмeчaниe к тeopeмe Д. Ф. Eгopoвa. D.A.N, 22, 1939, 309—311. [12] WESTON, J. D.: A counteг example conceгning Egoгoffs theoгem, J. London Math. Soc, 31, 1959, 139—140. [13] ZAKON, E.: On almost unifoгm conveгgence of families of functions. Cand. Math. Bull., 7,1964, 45—48. [14] ZAKON, E.: On "essentially metrizable" spaces and on measuгable functions with values in such spaces. Tгans. Ameг. Math. Soc, 119, 1965, 443—453. Received February 15, 1977. K a Ы г a matematickej analўzy Príгodovedeckej fakulty UK Mlynská dolina 816 31 Bгatislava ПOЧTЫ PABHOMEPHAЯ CXOДИMOCTЬ B CЛУЧAE HEПPEPЫBHЫX ПAPAMETPOB Tибop Hoйбpyн P.eзюмe Пycть (X, У, џ) пpocтpaнcтвo c впoлнe кoнeчнoň мepoй, Г-ceпapaбeльнoe топoлoгичecкoe пpocтpaнcтвo иcпoлняющee пepвyю aкcиoмy cчeтнocги. Beщecтвeннaя фyнкция / oпpeдeлeнa нa X x T oпpeдeляeт cиcтeмy фyнкций нa X. Имeннo, для tєT, /'(JC) -=/(JC, t). Ecли эти фyнкции измepимы и ecли \imf'(x) = <p(x) l—ÍO пoчти вcюдy, тo в нeкoтopыx cлyчaяx этa cxoдимocть џ — пoчти paвнoмepнa. Дoкaзывaeтcя, чтo oдним ycлoвиeм для џ — пoчти paвнoмepнoй cxoднимocти являeтcя квaзинeпpepыв-нocть x~ — ceчeний /,, фyнкции /. B paбoтe дaeтcя тaкжe дpyгoe ycлoвиe пoдoбнoгo типa. 328 

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Almost uniform convergence for continuous parameters

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Almost uniform convergence for continuous parameters

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