APPROXIMATING BOCHNER INTEGRALS BY RIEMANN SUMS

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1 APPROXIMATING BOCHNER INTEGRALS BY RIEMANN SUMS J.M.A.M. VAN NEERVEN Abstract. Let µ be a tight Borel measure on a metric space, let X be a Banach space, and let f : (, µ) X be Bochner integrable. We show that for every sequence of partitions P =,..., of satisfying N lim mesh `P = 0 there exists a sequence of sample point sets S = s,..., such that N N X Z lim µ` f`s f dµ = = 0. It is an old result of H. Lebesgue [5, pp. 30 ff.] that if f : [0, ] R is Lebesgue integrable, then there exist numbers [ n, n] such that n lim f ( ) b = f(t) dt. n = In this paper we will extend this result to Bochner integrable functions defined on an arbitrary metric space. Let be a metric space and let µ be a tight Borel measure on ; by definition, this is a finite Borel measure with the property that for every ε > 0 there exists a compact set K with µ( \ K) < ε. A partition of (, µ) is a finite collection P = {,..., N } of µ-measurable subsets of with the following properties: () µ( ) > 0 for all ; (2) µ( ) = 0 for all ; (3) µ( \ ) = 0. ( The numbers max µ( ) ) ( and max diam ( ) ) (whenever this is finite) will be called the measure of P and the mesh of P, respectively. A finite subset {s,..., s N } with s for each is called a set of sample points associated with the partition P. Let X be a Banach space. For a µ-measurable function f : X we define the Riemann sum of f relative to the partition P = {,..., N } and the associated sample point set S = {s,..., s N } by Our main result reads as follows. R(f; P, S) := a N µ( )f(s ). = 2000 Mathematics Subect Classification. Primary: 28B05; Secondary: 26A42, 46G0. Key words and phrases. Riemann sum, Bochner integral.

2 2 J.M.A.M. VAN NEERVEN Theorem. Let µ be a tight Borel measure on a metric space. Let X be a Banach space and let f : (, µ) X be a Bochner integrable function. Then for every sequence of partitions ( P ) of satisfying lim mesh (P ) = 0 there exists a sequence of associated sample point sets ( S ) such that lim R( f; P, S ) = f dµ strongly in X. Before starting with the proof we isolate some lemmas. The first is concerned with a topological property of tight measures on metric space. Lemma 2. Let µ be a tight Borel measure on a metric space and let X be a Banach space. Let f : X be µ-measurable. Then there exists a sequence (f n ) of bounded uniformly continuous functions on such that lim f n = f µ-almost everywhere. Proof. By definition of µ-measurability there exists a sequence of simple µ-measurable functions (φ n ) converging to f µ-almost everywhere. In particular for every m there is an index N m such that µ{φ Nm f} < m. Let us write φ Nm = K m =,m x,m with x,m X and with the sets,m pairwise disoint and µ-measurable. Suppose for the moment that we are able to find scalar-valued bounded uniformly continuous functions ψ,m such that µ{ψ,m,m } < mk m, =,..., K m. Then the functions ψ m = K m = ψ,m x,m are bounded and uniformly continuous, and we have µ{ψ m f} m + K m mk m = 2 m. Hence lim m ψ m = f in µ-measure, and the lemma follows by passing to a µ- almost everywhere convergent subseqence. This argument shows that it suffices to prove the following: For every µ-measurable set B there exists a sequence (ψ n ) of bounded uniformly continuous functions on such that lim ψ n = B in µ-measure. Choose a Borel subset C such that its symmetric difference with B satisfies µ(b C) = 0. Then B = C µ-almost everywhere. For each n, by [6, Theorem 3.] there is a compact subset C n C such that µ(c \ C n ) < n and a compact set n \ C with µ(( \ C) \ n ) < n. Define ψ n : [0, ] by ψ n (ω) := d(ω, n ) d(ω, n ) + d(ω, C n ). Then ψ n is bounded and uniformly continuous, and we have ψ n Cn n = C Cn n. Hence ψ n Cn n = B Cn n µ-almost everywhere. It follows that µ{ψ n B } µ( \ (C n n )) < 2 n and we conclude that lim ψ n = B in µ-measure.

3 APPROXIMATING BOCHNER INTEGRALS BY RIEMANN SUMS 3 Without loss of generality we will assume from this point onwards that µ() =. Lemma 3. Let f : (, µ) X be Bochner integrable. Let F G be µ- measurable sets such that µ(f ) > θµ(g) for some θ (0, ). Then there exists a point ω F such that µ(g) f(ω) f dµ. θ Proof. Suppose the contrary. Then for all ω F we have µ(g) f(ω) > f dµ. θ Integrating over the set F gives µ(g) f dµ θ µ(f ) a contradiction. F G G G f dµ > µ(g) f dµ, G Lemma 4. Let f : (, µ) X be Bochner integrable and let P = {,..., N } be a partition of. Suppose A and B are µ-measurable subsets of. Then there exists a set of sample points S = {s,..., s N } with the following properties: (i) µ 3µ(A); (ii) µ s S A s S B 3µ(B); (iii) For all s S (A B) we have () µ( ) f(s ) 3 f dµ. Proof. We choose the sample points s as follows. Case - If there exists s \ (A B), then pic such an s and put s := s. Suppose now such an s does not exist. Then we have A B. Case 2 - If µ( (A B)) > 3 µ( ), then by Lemma 3 there exists an s (A B) for which () holds. We tae such an s and put s := s. Suppose now that µ( (A B)) 3 µ( ). Case 3 - If there exists an s (B\A) for which () holds, then we pic such an s and put s := s. Case 4 - If there exists no s (B\A) verifying (), then by Lemma 3 we necessarily have µ( (B \A)) 3 µ( ). This implies µ( A) > 2 3 µ( ). Since we also have µ( (A B)) 3 µ( ) it follows that µ( (A\B)) > 3 µ( ). By Lemma 3 there exists an s (A\B) satisfying (). We choose such an s and put s := s. This rule defines a set of sample points S = {s,..., s N }.

4 4 J.M.A.M. VAN NEERVEN We can have s A only in the cases 2 and 4. In both cases we have µ( A) > 3 µ( ). Hence, µ = µ( ) 3 µ( A) 3µ(A). s S A s S A s S A Moreover, in both cases we have chosen s in such a way that () holds. We can have s B only in the cases 2 and 3. In both cases we have µ( B) > 3 µ( ). Hence, µ = µ( ) 3 µ( B) 3µ(B). s S B s S B s S B Moreover, in both cases we have chosen s in such a way that () holds. We now turn to the proof of Theorem. Step - First we prove: For every ε > 0 there exists an index N with the following property: for every n N there exists a sample point set S associated with P such that (2) R( f; P, S ) f dµ < ε. By absolute continuity, we can choose η > 0 so small that for all µ-measurable sets A with µ(a) < η we have (3) f dµ < 5 ε. For K define f K : X by { f(ω), if f(ω) K; f K (ω) := 0, else. A Define A K = { f > K}. By dominated convergence there exists K 0 large enough such that (4) f f K0 dµ < 5 ε and (5) 3µ(A K0 ) < η. For notational convenience we put g := f K0 and A := A K0. By Lemma 2 there exists a sequence (g ) of bounded uniformly continuous functions such that lim g = g µ-almost everywhere. Replacing each g by its truncation between K 0 and K 0, we may assume that sup sup g (ω) K 0. ω Define B = { g g > 0 ε}. By dominated convergence there exists an index 0 large enough such that (6) g g 0 dµ < 5 ε

5 APPROXIMATING BOCHNER INTEGRALS BY RIEMANN SUMS 5 and (7) µ(b 0 ) < 60K 0 ε. Again for notational convenience we put B := B 0 and h := g 0. Choose δ > 0 small enough such that (8) R(h; P, S) h dµ < 5 ε. whenever P is a partition of with mesh (P ) < δ and S is an associated sample point set. Such δ exists by the uniform continuity of h. From lim mesh (P ) = 0 we may choose N so large that mesh ( P ) < δ for all n N. For n N we apply Lemma 4 to the partitions P and the sets A and B and obtain sample point sets S verifying the conditions of the lemma. We fix n N and estimate: R( f; P, S ) f dµ R ( f; P, S ) R ( g; P, S ) R ( + g; P, S ) R ( h; P, S ) + R( h; P, S ) h dµ + h dµ g dµ + h dµ g dµ =: (I) + (II) + (III) + (IV) + (V). We will prove (2) by showing that each of these five terms is smaller than 5 ε. Distinguishing between points in A and \A respectively, noting that g ( ) = f ( ) when s A and g ( ) = 0 when s A, and using () we have (I) µ ( ) ( f ) ( ) s g s = S A S A µ ( ) f ( ) 3 ( Thans to (5) we have µ S A integral on the right hand side is less than 5 S S A f dµ. ) 3µ(A) < η and hence by (3) the ε. It follows that (I) 3 5 ε = 5 ε. Next, distinguishing between points S belonging to B and \ B respectively, noting that g(ω) K 0 and h(ω) K 0 for all ω, and using (7) we have (II) µ( ) ( ) ( ) g s h s + µ( ) ( ) ( ) g s h s S B 2K 0 µ S B + 0 ε µ 2K 0 3µ(B) + 0 ε < 2K 0 S B S B 3 60K 0 ε + 0 ε = 5 ε.

6 6 J.M.A.M. VAN NEERVEN The mesh of P being smaller than δ, by (8) we have (III) < 5ε. By (6) we have (IV) < 5 ε, and finally by (4) we have (V) < 5 ε. Step 2 - By Step, for every m there exists an index N m with the following property: for every n N m there exists a sample set S m associated with P such that R( f; P, S m ) f dµ < m, n N m. We may replace the numbers N m by larger ones and thereby assume that N < N 2 < N 3 <.... For n N we define S := S m, N m n < N m+ (m =, 2,... ) an for n < N we choose the sets S in an arbitrary way. The resulting sequence ( S ) has the desired properties. This concludes the proof of Theorem. With little extra effort we obtain the following stronger version of Theorem : Theorem 5. Let µ be a tight Borel measure on a metric space. Let X i be Banach spaces and let f i : (, µ) X i be Bochner integrable functions (i =, 2... ). Then for every sequence of partitions ( P ) of satisfying lim mesh ( P ) = 0 there exists a sequence of associated sample point sets ( S ) such that lim R( f i ; P, S ) = f i dµ (i =, 2,... ) strongly in X. Proof. By Theorem applied to the Banach space X X m and the function F m := (f,..., f m ), there exists an index N m with the following property: for every n N m there is a sample point set S m such that R( F m ; P, S m ) F m dµ < m, n N m. This immediately implies R( f i ; P, S m ) f i dµ < m, n N m (i =,..., m). The desired sequence of sample point sets ( S ) is finally obtained as in Step 2 of the proof of Theorem. Definition 6. Let f : X be an arbitrary function. We call a vector I X a Riemann sum limit of f if the following holds: for every sequence of partitions ( P ) with lim mesh (P ) = 0 there exists a sequence of sample point sets ( S ) such that lim N = µ ( ) ( ) f s I = 0. Using this terminology, Theorem states that if f : (, µ) X is Bochner integrable, then f dµ is a Riemann sum limit of f. If is totally bounded and f : (, µ) is essentially bounded, then f dµ is the unique Riemann sum limit of f; this follows from the following theorem.

7 APPROXIMATING BOCHNER INTEGRALS BY RIEMANN SUMS 7 Theorem 7. Let be totally bounded and let f : (, µ) X be an arbitrary function. Suppose x X is such that f, x is µ-integrable and µ-essentially bounded. If I is a Riemann sum limit of f, then I, x = f, x dµ. Proof. Let K > 0 be a constant such that the set N = { f, x > K } is µ-null. Fix n and divide the interval [ K, K] into disoint subintervals J,..., J M of diameter at most n. For m =,..., M let V m := {ω : f(ω) J m }. Since is assumed to be totally bounded it is possible to subdivide each V m of positive µ- measure into finitely many disoint subsets having positive µ-measure and diameter n. In this way we obtain a partition P = {,..., } N of of mesh n. Note that f(s), x f(t), x n for all s, t ; =,..., N. By assumption, associated to the resulting sequence ( P ) there exists a sequence of sample point sets ( T ) such that lim R( f; P, T ) = I. Then, lim R( f, x ; P, T ) = I, x. On the other hand, since f, x is µ-integrable, by Theorem there exists a sequence of sample point sets ( S ) such that lim R( f, x ; P, S ) = f, x dµ. We estimate: f, x dµ I, x f, x dµ R ( f, x ; P, S ) N + = µ ( ) f ( s ), x f ( t + R ( f, x ; P, T ) I, x f, x dµ R ( f, x ; P, S ) + R n + ( f, x ; P, T ) I, x. Passing to the limit n in the right hand side gives the desired result. ), x The following example shows that the converse of Theorem does not hold: a strongly measurable function with a Riemann sum limit need not be Bochner integrable, even if this limit is unique. Example 8. Let (e ) = denote the standard unit basis of the Hilbert space l2 and define f : (0, ] by f(t) := 2 e, t I := (2, 2 + ].

8 8 J.M.A.M. VAN NEERVEN Clearly f is strongly measurable and f(t) dt = (0,] = I 2 = = =. Thus f fails to be Bochner integrable. Next we will show that I := = e is a Riemann sum limit integral of f. To this end let ( P ) be an arbitrary sequence of partitions of (0, ] satisfying lim mesh (P ) = 0. Fix ε > 0 arbitrary and fix an integer K such that 2 K < 2 ε and =K+ < 2 4 ε2. Choose n 0 so large that mesh (P ) 2 K for all n n 0. Fix an index n n 0 and write P =: P = {,..., N }. We define a sample point set S =: S = {s,..., s N } by the following rule: for =,..., N let () = min{ : I } and let s be an arbitrary point in I (). For let J be the set of all {,..., N} for which we have s I. From diam ( ) < 2 K and I m = (m =,..., ) we have, for all =,..., K, (2, K ] J (2 2 K, 2 + ]. For these it follows that 2 2 K Hence, f(s ) e J Summing over =,..., K we obtain K K f(s ) = J = J K. = 2 e e J = 2 J 2 K+. e K = 2 K+ 2 K. Next fix K +. If s I, then I m = (m =,..., ) and therefore J (0, 2 + ]. Put Noting that β := J. J 2

9 APPROXIMATING BOCHNER INTEGRALS BY RIEMANN SUMS 9 we have Hence, =K+ J f(s ) β m = 2 J. 2 e = =K+ =K+ (β ) e 2 =K+ Note that in the double sum on the left hand side only finitely many terms are non-zero. Putting everything toghether we obtain N N f(s ) I = f(s ) = e = = K f(s ) e + f(s ) = J =K+ e =K+ J =K+ ( ) 2 2 K + 2 < 2 ε + 2 ε = ε. =K+ This shows that I = = e is a Riemann sum limit of f. The uniqueness of I as Riemann sum limit of f follows from Theorem 7 applied to the coordinate functionals e. To conclude we compare our results with two theories of generalized Riemann integration: the theories of McShane integration and Henstoc-Kurzweil integration. For simplicity we confine the discussion to the case where is the unit interval [0, ]. For more details we refer to the boo [4] and the papers [], [3]. Let P be a partition of [0, ] and let S be a finite subset of [0, ]. Let us call a pair (P, S) a McShane pair if P is of the form P = {[p, p ] : =,..., N} with 0 = p 0 <... p N = and if S = {s,..., s N } [0, ]. A gauge is a strictly positive function δ on [0, ]. We call the McShane pair (P, S) subordinate to the gauge δ if [ p, p ] ( s δ(s ), s + δ(s ) ), =,... N. A function f : [0, ] X is called McShane-integrable, with integral I X, if for every ε > 0 there exists a gauge δ such that R(f; P, S) I < ε for every McShane pair (P, S) subordinate to δ. Here, of course, R(f; P, S) := N (p p ) f(s ). = By a Henstoc-Kurzweil pair we mean a McShane pair (P, S) with the property that S is a sample point set for P, i.e. we have s [p, p ] for all. A function 2.

10 0 J.M.A.M. VAN NEERVEN f : [0, ] X is called Henstoc-Kurzweil-integrable, with integral I X, if for every ε > 0 there exists a gauge δ such that (9) R(f; P, S) I < ε for every Henstoc-Kurzweil pair (P, S) subordinate to δ. Clearly every McShane integrable function is Henstoc-Kurzweil integrable, with the same integral I. The following is proved in [3, Theorem 6]: Theorem 9. Every Bochner integrable function f : [a, b] X is McShane integrable, and therefore Henstoc-Kurzweil integrable, and the integrals coincide. Let us compare this result with Theorem in the case = [a, b]. By taing ε = n in (9), from Theorem 9 it follows immediately that the integral of a Bochner integrable f : [a, b] R can be realized as the limit of certain Riemann sums. The point of Theorem, however, is that partitions allowed there need not be subordinate to the gauges used in (9). In the converse direction, the existence of a Riemann sum limit for f in the sense of Definition 6 does not produce a sequence of gauges in any obvious way, which leaves open the possibility that such a function fails to be Henstoc-Kurzweil integrable. Let us point out in this connection that by [3, Theorem 5], the function f in Example 8 is indeed McShane integrable, and hence Henstoc-Kurzweil integrable; cf. [, Example 3E]. Finally we mention the paper [2], where McShane- and Henstoc-Kurzweil integrability of vector-valued functions on metric spaces is studied. In contract to the approach ust described, in this paper the partitions of consist of countably many sets. Acnowlegdement - The author is indebted to Anton Schep for pointing out the references [] and [2]. References [] D.H. Fremlin and J Mendoza, On the integration of vector-valued functions, Illinois J. Math. 38 (994), [2] D.H. Fremlin, The generalized McShane integral, Illinois J. Math. 39 (995), [3] R.A. Gordon, The McShane integral of Banach-valued functions, Illinois J. Math. 34 (990), [4] R. Henstoc, Theory of Integration, Butterworths, London, 963. [5] H. Lebesgue, Sur les integrales singulières, Ann. Fac. Sci. Univ. Toulouse (909), [6] K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, 967. Department of Applied Mathematical Analysis, Technical University of Delft, P.O. Box 503, 2600 GA Delft, The Netherlands address: J.vanNeerven@its.tudelft.nl