Appendix Nonatomic Measure Spaces

In this appendix we collect some facts about finite measure spaces without (and with) atoms, which are applied in Chapter 7. Let (Ω, , µ) be a σ-finite measure space. Then an atom of µ is a set A with Sµ(A) > 0 such that for all C with C A, either µ(C) = 0 or µ(C∈) S = µ(A). By σ-finiteness, we have µ∈(A S) < + .(⊂Ω, , µ) or µ is called nonatomic if it has no atoms. ∞ S Proposition A.1. Let (Ω, , µ) be a nonatomic finite measure space with µ(Ω) > 0. Then for any cSwith 0 < c < µ(Ω), there is an A with µ(A) = c. ∈ S

Proof. We can assume that µ(Ω) = 1. It will first be shown that for some C , 1/3 µ(C) 2/3. Suppose not. Let p := sup µ(B): µ(B) < 1/3 1/3.∈ SThen p≤ > 0 since≤Ω is not an atomand if 2/3 < µ({D) < 1, let B := Ω }D ≤. Take B with µ(B ) p. Let E := n B . It will be shown by induction\ n ∈ S n ↑ n j=1 j that µ(En) < 1/3 for all n. This is true for n = 1. Assuming it holds for a given n, we have µ(E ) = µ(E BS ) < (1/3)+(1/3) = 2/3. Thus by the n+1 n ∪ n+1 assumption that µ takes no values in [1/3, 2/3], µ(En+1) < 1/3, completing ∞ the induction. Let E := n=1 Bn. Then µ(E) = p, so p < 1/3. Now, Ω E is not an atom, so take F Ω E with 0 < µ(F ) < 1 p. If µ(F ) (2/3)\ p, then pS < µ(E F ) 2/⊂3, contradicting\ the assumption− or the definition≤ − of p. But if µ(F ) >∪(2/3)≤ p, replacing F by Ω (E F ) leads to the same contradiction. So, it is shown− that for some C ,\ 1/3∪ µ(C) 2/3. ∈ S ≤ ≤ n Next, an inclusion-chain will be a finite sequence Cj j=0 with = n { } ⊂ S ∅ C0 C1 Cn = Ω. One inclusion-chain Cj j=0 will be a refinement ⊂ ⊂ · · · ⊂k { } of another Di i=0 iff for each i, Di = Cj for some j. Let 0 := ,Ω . Given { } n 2n C {∅ } an inclusion-chain n = Cj j=0, define a refinement n+1 of it by adjoining, C n { } C n n for each j = 1,..., 2 , a set Cj−1 Aj where Aj , Aj Cj Cj−1, and n n ∪ ∈ S ⊂ \ µ(Aj )/µ(Cj Cj−1) [1/3, 2/3]. Thus n are defined recursively for all n. \ ∈ C n Clearly, for each n and j = 1,..., 2n, if = Cn 2 , then µ(Cn Cn ) Cn { j }j=0 j \ j−1 ≤ R.M. Dudley and R. Norvaiša, Concrete Functional Calculus, Springer Monographs 645 in Mathematics, DOI 10.1007/978-1-4419-6950-7, © Springer Science+Business Media, LLC 2011 646 Appendix Nonatomic Measure Spaces

n n 2n (2/3) 0 as n . Thus the values µ(Cj ) j=0 become dense in [0, 1] as → → ∞ { n } n . For any t [0, 1], let B := 2 Cn : µ(Cn) t . Then B → ∞ ∈ t n j=0{ j j ≤ } t ∈ S and µ(Bt) = t, completing the proof. 2 S S

Proposition A.1 and induction on n give directly the following:

Corollary A.2. Let (Ω, , µ) be a finite nonatomic measure space. Let ri S n for i = 1, . . . , n be numbers with ri > 0 and i=1 ri = µ(Ω). Then Ω can be decomposed as a union of disjoint sets Ai with µ(Ai) = ri for i = 1, . . . , n. ∈ SP Here are sufficient conditions under which atoms equal singletons up to sets of measure 0, as holds in familiar cases. A measurable space (Ω, ) is called separated iff for every x = y in Ω, there is some C containingS just one of x and y. 6 ∈ S Proposition A.3. Let (Ω, , µ) be a σ-finite measure space such that (Ω, ) is separated and is countablyS generated, i.e. for some countably many A S S j ∈ , is the smallest σ-algebra containing all Aj. Then for any atom A of µ thereS S is an x A such that µ(A x ) = 0. So, the singleton x is also an atom. This∈ holds in particular\{ if there} is a metric on Ω for{ which} it is separable and is its Borel σ-algebra. S Proof. Let A be an atom of µ. By σ-finiteness we have 0 < µ(A) < + . For each j, let B := A if µ(A A ) = µ(A) or B := Ac if µ(A A ) =∞ 0. Let j j ∩ j j j ∩ j C := A ∞ B . Then µ(C) = µ(A) and C is also an atom. If there exist ∩ j=1 j x = y in C then by the separated assumption there must exist Aj such that 6 T Aj contains one of x and y but not both, contradicting the definition of C. Thus C is a singleton x and µ(A x ) = 0 as stated. The Borel σ-algebra{ of} a separable\{ metric} space is generated by the countably many open balls with centers in a countable dense set and rational radii, and it clearly separates points. This completes the proof. 2

0 Let λ = λ(Ω, , µ) be the set of all functions G (Ω, , µ) such that µ G−1 Dhas aD boundedS density ξ with respect to Lebesgue∈ L measureS λ on R. ◦ G Such functions were characterized in Theorem 7.24. If the set λ(Ω, , µ) is nonempty then µ must be nonatomic as the following shows: D S Proposition A.4. Let (Ω, , µ) be a ﬁnite measure space and G 0(Ω, , µ). If µ has an atom then µ G−S1 has an atom y for some y. ∈L S ◦ { } Proof. Suppose that A is an atom of µ with c := µ(A) > 0. Restricting µ to A, we can assume∈ that S Ω = A. Let F (x) := µ(G−1(( , x])) [0, c] for each x R. Then F (x) 0 as x and F (x) −∞c as x ∈ + . If F (x) (0∈, c) for some x ↓ R, then→ the −∞ set G−1(( ↑, x]) →gives∞ a contradiction∈ to the fact that∈A is an atom. Thus F has−∞ only∈ one S point of Appendix Nonatomic Measure Spaces 647 increase, say y R, with F (x) = 0 for all x < y and F (x) = c for all x y, and so y is an∈ atom of µ G−1. This completes the proof. ≥2 { } ◦

On the other hand if (Ω, , µ) is nonatomic, then the set λ(Ω, , µ) is rich enough in the sense statedS by the following two propositions.D S Proposition A.5. For any nonatomic ﬁnite measure space (Ω, , µ) with S µ(Ω) > 0 and < a < b < , there is a G λ such that the density ξ equals c1−∞, where c = µ(∞Ω)/(b a). ∈ D G [a,b] − Proof. If G exists such that the density ξG equals µ(Ω)1[0,1], then a+(b a)G satisﬁes the conclusion. So we can assume a = 0 and b = 1. Also, multiplying− µ by a constant, we can assume µ(Ω) = 1. By Proposition A.1, take A with µ(A ) = 1/2. Let A := Ω A , 11 ∈ S 11 12 \ 11 and G1 := (1/2)1A12 . At the jth stage we will have a decomposition of Ω j j j into 2 disjoint sets Ajk , k = 1,..., 2 , with µ(Ajk) = 1/2 and Aj,k = ∈ S j j Aj+1,2k−1 Aj+1,2k for each j 1 and k = 1,..., 2 . Let Gj := (k 1)/2 on ∪ j ≥ −1 − Ajk for k = 1,..., 2 . Then µ Gj is uniformly distributed over the points j j ◦ (k 1)/2 for k = 1,..., 2 . As j , Gj converges uniformly to a µ- − → ∞ −1 −1 measurable function G. So the probability laws µ Gj converge to µ G ◦ ◦ 2 with the density ξG = 1[0,1], completing the proof of the proposition.

Proposition A.6. For any nonatomic ﬁnite measure space (Ω, , µ) and 1 s < , bounded elements of are dense in Ls(Ω, , µ). S ≤ ∞ Dλ S k Proof. We know that µ-simple functions h := i=1 ci1Ai are dense. By Propo- sition A.5, for any δ > 0 and i = 1, . . . , k, there is a µ-measurable function P −1 fi on Ai such that for the restriction µi of µ to Ai, µi fi has density ◦ k µ(Ai)1[ci,ci+δ]/δ with respect to Lebesgue measure. Let G := i=1 fi1Ai . Then G h δµ(Ω)1/s and µ G−1 has density ξ µ(Ω)/δ < . The s G P proof ofk the− propositionk ≤ is complete.◦ ≤ ∞ 2

The following is used to prove the necessity of a suitable measurability assumption on Nemytskii operators acting on Lp spaces (Theorem 7.13(c)). Proposition A.7. Let (Ω, , µ) be any nonatomic probability space and let Q be any probability measureS on the Borel sets of R. Then there is a G 0(Ω, , µ) such that µ G−1 = Q. ∈ L S ◦ Proof. By Proposition A.5, take H 0(Ω, , µ) such that µ H−1 = U[0, 1] (Lebesgue measure restricted to [0, 1]).∈ L Let FS(x) := Q(( , x◦]) for all x R and F ←(y) := inf x: F (x) y for 0 < y < 1. Then U[0−∞, 1] (F ←)−1 = Q∈[53, Proposition 9.1.2].{ Since F ←≥is} Borel measurable, G := F ←◦ H 0(Ω, , µ) and µ G−1 = Q. ◦ ∈ L S 2 ◦ 648 Appendix Nonatomic Measure Spaces

The following is used to show that continuity of a Nemytskii operator implies its boundedness and that several smoothness properties of Nemytskii operators valid on a nonempty open subset of Ls can be extended to all of Ls. Proposition A.8. Let (Ω, , µ) be a nonatomic σ-finite measure space. Let 1 S g (Ω, , µ), g 0, and g dµ > 0. For each E , let ν(E) := E g dµ. Then∈ L (Ω, S, ν) is a≥ nonatomic∫ finite measure space.∈ S ∫ S Proof. Clearly ν is a finite measure. Suppose that A is an atom of ν. Then ν(A) > 0 and µ(A) > 0. Let B := ω A: g(ω) = 0 . Then µ(B) > 0. Take C such that C B and 0 <{ µ(C∈) < µ(B).6 Then} both ν(C) > 0 and ν(A∈ SC) = ν(B C)⊂> 0, a contradiction, proving that ν is nonatomic. 2 \ \

Proposition A.9. Let (Ω, , µ) be a nonatomic ﬁnite measure space with µ(Ω) > 0, and let 1 s < S. For each c > 0 and h Ls(Ω, , µ) such that ≤ ∞ n ∈ S h s > c there exists a partition Ai i=0 of Ω into measurable sets such that k k h s dµ = h s/(n + 1) (cs/2{, cs}] for i = 0, . . . , n. ∫Ai | | k ks ∈ Proof. Let g := h s and apply Proposition A.8. Then (Ω, , ν) is nonatomic. | | s sS s Let n be the unique positive integer such that nc < h s (n + 1)c . Then the desired partition A n of Ω exists by Corollaryk A.2.k ≤ 2 { i}i=0

Notes. The statement of Proposition A.1 is [53, 3.5 Problem 11] and [87, 41 Problem 2]. § § Regarding Proposition A.3, it is known, e.g. Cohn [34, Corollary 8.6.4], that if a measurable space (Ω, ) is separated and is countably generated then there exists a metric d on SΩ for which it is separableS and is the Borel σ-algebra. Thus the “particular” case is not really special. S References

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243. L. C. Young. The Theory of Integration. Cambridge University Press, 1927. Repr. Stechert-Hafner, New York, 1964. 244. L. C. Young. An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. (Sweden) 67 (1936), 251–282. 245. L. C. Young. Inequalities connected with bounded p-th power variation in the Wiener sense and with integrated Lipschitz conditions. Proc. London Math. Soc. (Ser. 2) 43 (1937), 449–467. 246. L. C. Young. Sur une généralisation de la notion de variation de puissance pième bornée au sens de M. Wiener, et sur la convergence des séries de Fourier. C. R. Acad. Sc. Paris 204 (1937), 470–472. 247. L. C. Young. General inequalities for Stieltjes integrals and the convergence of Fourier series. Math. Ann. 115 (1938), 581–612. 248. L. C. Young. Some new stochastic integrals and Stieltjes integrals. Part I. Analogues of Hardy-Littlewood classes. In: Advances in Probability and Related Topics, Vol. II (P. Ney, ed.), 161–240. Dekker, New York, 1970. 249. L. C. Young. Some new stochastic integrals and Stieltjes integrals. Part II. In: Advances in Probability and Related Topics, Vol. III (P. Ney, ed.), 101–178. Dekker, New York, 1974. 250. L. C. Young. A converse to some inequalities and approximations in the theory of Stieltjes and stochastic integrals, and for nth derivatives. Studia Math. 55 (1976), 215–223. 251. R. C. Young. On Riemann integration with respect to a continuous increment. Math. Ann. 29 (1928), 217–233. 252. W. H. Young. The Fundamental Theorems of the Differential Calculus. Cam- bridge University Press, 1910. Repr. Hafner, New York, 1971. 253. W. H. Young. On classes of summable functions and their Fourier series. Proc. Roy. Soc. London (Ser. A) 87 (1912), 225–229. 254. W. H. Young. On integration with respect to a function of bounded variation. Proc. London Math. Soc. (Ser. 2) 13 (1914), 109–150. 255. W. H. Young. On a formula for an area. Proc. London Math. Soc. (Ser. 2) 18 (1919), 339–345. 256. A. C. Zaanen. Riesz Spaces II. North-Holland, Amsterdam, 1983. 257. E. Zeidler. Nonlinear Functional Analysis and its Applications. I: Fixed-Point Theorems. Springer-Verlag, New-York, 1986. 258. A. Zygmund. Trigonometric series. 2d ed., Cambridge University Press, 1959, repr. with corrections and some additions 1968, repr. 1988.

An asterisk (∗) indicates works we have seen cited in secondary sources but have not seen in the original. Subject Index

Page numbers in italics indicate pages with the main or most general definition of each term additive transform, 438 chain rule, 1, 85, 237, 631 analytic continuation, 258–260 character, 222 analytic mapping, 258 characteristic exponent, 612 area of curve, 632 common discontinuities, 25 α-stable Lévy motion, 615 common one-sided discontinuities, 25 complementary function, 110, 109–114 B-Taylor series, 246 complexification, 232, 231–233 B-analytic function, 246 component, 96 B-power series, 246 composition, 237, 393, 393–406 Banach algebras, 218, 215–235 concave function, 15 natural, 282 conditional probability, 600 semisimple, 283 connected, 91 Banach fixed point theorem, 518 locally, 96 Banach space of functions, 11 simply, 96 Beurling’s condition, 168 contraction, 518 bimeasurable function, 335 convergence Borel space, 600 in µ-measure, 15 bounded k-linear operator, 239 bounded bilinear operator, 9 in outer µ-semivariation, 43 bounded linear operator, 10 unconditional, of a product, 417 bounded semivariation, 26, 41 unconditional, of a sum, 13 Brownian bridge, 618, 618–628, 632 convex function, 15 Brownian motion, 575, 575–576, convex set, 14 629–640 convolution of interval functions, 196 convolution of point functions, 198 Ck function, 255 convolution semigroup, 612 C∞ function, 255 continuous, 612 Carathéodory function, 341 curve, 90 Cauchy–Hadamard formula, 244 null-homotopic, 96 chain, 226 reversed, 225 662 Subject Index degree of lacunarity, 149 hereditarily S-measurable set, 336 derivative, of a nonlinear operator, 237 Hölder condition of order α, 4, 11 kth, 252 locally, 274 differentiable in ν-measure, 357 locally s-uniformly, 301 differential equations, 7–9 holomorphic function, 91 formal linear, 496 Hurst index, 589 formal nonlinear, 507 linear, 496–499 index of p-variation, 588 nonlinear, 505–549 infinitely divisible law, 612 differential, kth, 252 integral equations, 7–9 Dirichlet kernel, 555 backward homogeneous linear, 477 Duhamel’s formula, 452, 451–456 backward nonlinear, 507 ∆2 condition, 103 forward homogeneous linear, 474 forward nonlinear, 507 empirical distribution function, 616 inhomogeneous linear, 478 empirical process, 616, 615–627 linear, 7, 474–496 essentially bounded, 14 nonlinear, 505–549 exponential Fourier series, 551 integral transform, 197 integrals filtration, 573 Bartle, 43, 41–46, 100 generated, 573 Bochner, 39, 38–41, 499 satisfying the usual hypotheses, 576 central Young, 55, 51–56 finite-dimensional joint distributions, contour, 92 571 Darboux–Stieltjes, 98 fixed discontinuity, 572 full Stieltjes, 47, 159 fixed point, 518 Henstock–Kurzweil, 57, 56–71 Fourier series, 552, 551–569 Itô, 630 convergent uniformly, 555 Kolmogorov, 6, 32 fractional Brownian motion, 589 Lebesgue–Stieltjes, 4 functional calculus, 229 McShane, 71 Φ-variation (for point functions), 103, mean Stieltjes, 631 103–116, 127–138 refinement Riemann–Stieltjes, 2, 24 norm, 105 refinement Young–Stieltjes, 5, 29 seminorm, 105 relations between, 87–90 sum, 103 Riemann–Stieltjes, 2, 24, 24–28 Φ-variation (for interval functions), 117, Ross–Darboux–Stieltjes, 100 116–127, 138–142 stochastic, 629–631, 640 norm, 118 Ward–Perron–Stieltjes, 63 seminorm, 118 with two integrands, 442–451 sum, 117 Y1, Y2, 51 Φ-variation stochastic process, 573 integration by parts, 78–81 interval, 11 Gaussian probability measure, 580 nondegenerate, 11 Gaussian stochastic processes, 579, interval function, 6, 19 579–599 additive, 6, 19 gauge function, 56 bounded, 410 Gelfand topology, 224 invertible, 413 Gelfand transform, 224 multiplicative, 409, 408–415 growth condition, 338 nondegenerate, 412 Subject Index 663

reciprocal, 413 Nemytskii operator, 273 upper continuous, 6, 19 autonomous, 273 upper continuous at ∅, 19 nonatomic measure space, 338, 645 interval partition, 6 normal probability measure, 580 associated Young, 34 refinement of, 6 operator tagged, 6 bounded, 293 tagged refinement of, 6 (Fréchet) differentiable, 237, 628 tags of, 6 differentiable of order k, 252 trace Young, 34 k-linear, 239 Young, 6, 12 multiplication, 275 invertible function, 409 Nemytskii, 273, 273–333 Itôformula, 630 product integral, 457 Itôisometry, 630 two-function composition, 393 operator norm, 10 k-linear operator, 107 Orlicz norm, 114 M-bounded, 107 Orlicz space, 14 KMT construction, 624 oscillation, 10, 604 Kolmogorov sum, 6, 32 perfect measure space, 339 Lévy measure, 612 Picard iterates Lévy process, 613, 611–615 backward, 511 Lipschitz property, 299 forward, 510 locally, 299 point partition, 2, 12 locally bounded Φ-variation, 573 δ-fine tagged, 56 locally bounded p-variation, 573 mesh of, 12 Love–Young inequalities, 173–188, mesh of tagged, 2, 12 444–451 refinement of, 2, 12 tagged, 2, 12 major function, 62 tagged intervals of, 12 Markov kernel, 601 tagged refinement of, 2, 12 Markov process, 601, 600–611 Young tagged, 5, 12 Markov property, 602 polarization, 241 martingale, 576, 576–579 polynomial, between Banach spaces, local, 577 242, 239–251 mean value theorem, 238 k-homogeneous, 241 measurable space, 600 Taylor, 265 minor function, 62 power series in Banach spaces, 243 modification of stochastic process, 571 probability kernel, 600 modulus of continuity, 115, 115–116 product integral, 9, 408, 407–503 multiplicative transform, 432 strict, 433 µ-almost separably valued function, 38 p-variation (for interval functions), 13, µ-measurable function, 14 117 µ-null set, 43 norm, 118 µ-(pre)-ideal space, 398 seminorm, 118 full, 398 p-variation (for point functions), 3, 13 µ-simple function for an additive µ, 42 local, 142 µ-simple function for a subadditive µ, norm, 3, 13 38 seminorm, 3, 13 664 Subject Index

sum, 3, 13 strong, 336 p-variation stochastic process, 573 solution mapping, 483, 536 p∗-variation spectral radius, 219, 219–222, 281 for interval functions, 124 spectrum, 221 for point functions, 142 stationary increments, 575 step function, 18 quadratic covariation, 631 Stieltjes pair, 159 quadratic variation, 631 stochastic continuity, 572 quantile function, 616 stochastic discontinuity, 572 stochastic process, 571, 600 radius of uniform convergence, 244 adapted, 573 reciprocal function, 409 cadlag, 571 regular conditional distribution, 601 ladlag, 571 regulated function, 5, 17 separable, 573 regulated sample function, 571 stopping time, 577 remainder in differentiation, 237 substitution rule, 83, 188, 233 remainder in Taylor’s expansion, 268 sup norm, 10 resolvent, 221 symmetric function, 240 Riemann–Stieltjes sum, 2, 24 symmetrization, 241 s-uniformly α-Hölder, 301 Taylor series in Banach spaces, 91, 258, sample function, 571 251–269 Schwarz theorem, 253 Taylor’s formulas, 264–269 semimartingale, 577 tensor product, 269, 269–270 semivariation of a function, 26 separated measurable space, 646 u-differentiable, 277 separating set, 574 on the graph, 277 sequence uniformly entire mapping, 261 D’yaˇckov, 151 universally measurable, 336 Hadamard-lacunary, 149 W (φ, ψ)-, 150 winding number, 90, 226, 632 W ′(φ, ψ)-, 150 ξ-lacunary, 149 Young, W. H., inequality, 110 Shragin function, 336 Young–Stieltjes sum, 5, 29 Author Index

Ahlfors, L. V., 93, 102 de Finetti, B., 102 Alexiewicz, A., 271 Dellacherie, C., 643 Appell, J., 273, 324, 333, 390, 391, 406 Diestel, J., 100 Artin, E., 102 Dieudonné, J., 270, 271, 406 Dobrushin, R. L., 500, 501 Baernstein, A., 566, 569 Doléans-Dade, C., 643 Banach, S., 234, 235, 270, 549 Dollard, J. D., 500, 502 Bartle, R. G., 43, 100 Doob, J. L., 572, 574, 640 Bary, N. K., 553 Doran, R. S., 234 Beardon, A. F., 97 Dudley, R. M., VIII, 101, 618, 642 Belfi, V. A., 234 Duncan, J., 235 Beurling, A., 168 Dunford, N., 100, 234 Birkhoff, G., 100, 499 Durdil, J., 391 Birnbaum, Z., 211 Dvoretzky, A., 618 Blumenthal, R. M., 614, 615 Dynkin, E. B., 641 Bochnak, J., 270 Bochner, S., 100 Ebin, D. G., 406 du Bois-Reymond, P., 557 Edwards, R. E., 551, 552, 569 Bonsall, F. F., 235 Bretagnolle, J., 614, 615, 625, 643 Fernholz, L. T., 406 Brokate, M., 406 Fernique, X., 590 Fisk, D. L., 578 Carleman, T., 553 Fréchet, M., 270 Carleson, L., 552 Freedman, M. A., 500, 501 Cena, A., 249, 271 Friedman, C. N., 500, 502 Chae, S. B., 270, 271 Fristedt, B., 642 Ciemnoczo lowski, J., 333 Cohn, D. L., 648 Garay, B. M., 406 Colonius, F., 406 Gehring, F. W., 214 Gelfand, I. M., 222, 234, 235 D’yaˇckov, A. M., 212 Getoor, R. K., 614, 615 Dalecki˘ı, Ju. L., 497–499, 503 Gihman, I. I., 614, 641 Dales, H. G., 229, 333 Gill, R. D., 500, 501 Daniell, P. J., 99 Gochman, E. Ch., 99 666 AuthorIndex

Golubov, B. I., 569 Lévy, P., 613, 640, 643 Gordon, R. A., 62, 101 Lipschitz, R., 568 Gray, A., 406 Lojasiewicz, J., 270 Greene, R. E., 228 Love, E. R., 102, 212, 213 Luxemburg, W. A. J., 211, 406 Hartung, F., 406 Lyons, T., 501, 549, 643 Hausdorff, F., 234 Henstock, R., 101 MacNerney, J. S., 500, 502 Hewitt, E., 102 Major, P., 624, 642 Hildebrandt, T. H., 99, 100, 102, 500, Manhas, J. S., 394 502, 631 Manstaviˇcius, M., 603, 615, 640, 641 Hönig, C. S., 502 Marcinkiewicz, J., 211, 214, 553, 557, Hostinsky, B., 499 569 Huang, Y.-C., 618, 642 Marcus, M. B., 590 Hunt, R. A., 552 Marsden, J., 406 Masani, P. R., 499 Ince, E. L., 549 Massart, P., 618, 625, 643 Itô, K., 575, 613, 630 Matkowski, J., 321, 333 Mazur, A., 270 Jacob, M., 102 Mazur, S., 222, 234 Jain, N. C., 640 McKean, H. P., 575, 630, 643 Jarn´ık, J., 500 McLeod, R. M., 101 Johansen, S., 500, 501 McShane, E. J., 71, 101 Medvedev, F. A., 99 Kallenberg, O., 571, 602, 603, 613, 631, Meyer, P.-A., 643 640, 641 Milnor, J., 406 Katznelson, Y., 552 Mi´s, J., 321, 333 Kiefer, J., 618 Monrad, D., 640 Kinney, J. R., 641 Monroe, I., 577, 614 Kirwan, P., 235 Moore, E. H., 99 Kluvanek, I., 212 Musielak, J., 211, 243 Kolmogorov, A. N., 6, 552, 571, 613, Neumann, J. von, 272 618 Newman, M. H. A., 97 Komlós, J., 642 Nguyen, T., 271 König, G., 98 Nisio, M., 640 Kopeć, J., 243 Norvaiˇsa, R., VIII Krabbe, G., 211 Krantz, S. G., 228, 552 Oodaira, H., 640 Krasnosel’ski˘ı, M. A., 390 Orlicz, W., 211, 213, 214, 270, 271, 333, Kre˘ın, M. G., 497–499, 503 553 Kurzweil, J., 101, 500 Peano, G., 407 Lai, T. L., 640 Peng-Yee, L., 101 Landau, H. J., 590 Pfeffer, W. F., 71 Larsen, R., 234 Picard, E., 549 Le´sniewicz, R., 213, 214 Pollard, S., 74, 98, 99 Lebesgue, H., 99, 552 Ponosov, A. V., 390 Ledoux, M., 584, 591, 640 Lehmann, E. L., 368 Qian, J., 616, 618, 642 AuthorIndex 667

Radon, J., 99 Stevenson, J. O., 406 Raikov, D., 234 Stieltjes, T. J., 98 Rasch, G., 499 Szénássy, B., 98 Reeds, J. A., 406 Riesz, F., 98, 343 Talagrand, M., 591 Riesz, M., 552 Taylor, A. E., 270, 271 Rosenthal, P., 394 Taylor, S. J., 599, 642 Ross, K. A., 99 Turi, J., 406 Rubinˇste˘ın, A. I., 212 Tusnády, J., 642 Runst, T., 391 Tvrdý, M., 502 Ruston, A. F., 235 Uhl, J. J., 100 Sainte-Beuve, M.-F., 337, 343 Va˘ınberg, M. M., 390, 391 Saks, S., 71, 101 Vejvoda, O., 502 Salem, R., 569 Volterra, V., 407, 499 Schatten, R., 271, 272 Schlesinger, L., 499 Wang Sheng-Wang, 391 Schmidt, G., 499 Ward, A. J., 71, 101 Schoenberg, I. J., 589 ˇ Whittle, P., 642 Schwabik, S., 102, 500, 502 Wiener, N., 211, 212, 557, 569, 640 Schwartz, J. T., 100, 234 Wolfowitz, J., 618 Shakarchi, R., 228, 230 Shapiro, J. H., 394 Xu, Q., 642 Shepp, L. A., 590 Shilov, G., 234 Young, L. C., 51, 99, 101, 173, 211–214, Shragin, I. V., 390 569 Siciak, J., 270 Young, R. C., 99 Sickel, W., 391 Young, W. H., 5, 99, 211, 271, 632, 643 Singh, R. K., 394 Skorohod, A. V., 614, 641, 643 Zaanen, A. C., 406 Slepian, D., 591 Zabrejko, P. P., 273, 324, 333, 390, 391, Smirnov, N. V., 618 406 Smith, H. L., 99, 101 Zeidler, E., 271 Stein, E. M., 228, 230 Zygmund, A., 214 Index of Notation

← · , bounded bilinear operator, 9 kµk(p), 1 ≤ p < ∞, 436 ← ≺ , A ≺ B means s < t for all s ∈ A kµk(p), 1 ≤ p < ∞, 437 and t ∈ B, 12 kµk[p],W , 486 ⋍, equivalence of bimeasurable kµk← , 493 functions, 337 [p],W kµk[Φ], 118 ⊏, refinement of partitions, 121 kµk(Φ), 118 [[·, ·]], general interval, 12 =R A f·dµ, Kolmogorov integral, 32 [[·, v), interval right open at v, 12 b =R f·dh, 36 [[·, v], interval right closed at v, 12 a (u, ·]], interval left open at u, 12 AI, additive, upper continuous, 21 [u, ·]], interval left closed at u, 12 AIp, of bounded p-variation, 117 ∗ ∗ (·, ·), open interval, 11 AIp, having p -variation, 124 [·, ·), left closed, right open, 11 AIΦ, of bounded Φ-variation, 117 (·, ·], left open, right closed, 11 AI , vector space, 119 f Φ [·, ·], closed interval, 11 αn = βn◦F , empirical process, 616 k · k(Hα), α-Hölder seminorm, 11 B R B k · kHα , α-Hölder norm, 219 R, -valued functions in , 290 [U] k · k∞, essential supremum, 14 B , closure of range is in U, 287 k · kp, 1 ≤ p < ∞, 14 B2/B1, 401 k · ksup, supremum, 10 B = {Bt}t≥0, Brownian motion, 575 ◦ ◦ k · k1+α,∞, 404 B = {Bt }0≤t≤1, bridge, 618 B(·, ·), bounded bilinear operator, 9 k · k2/1, 401 Bo, Bochner, 38 kfk[p], 1 ≤ p < ∞, 3, 13 kfk , 1 ≤ p < ∞, 3, 13 (p) CRG = UCR ∩ UG , 508 kfk , Luxemburg norm, 15 β β Φ CV, 15 |||f|||Φ , 113, 563 CV∗, 110 |||f|||(Φ) , 112 CWp, 137 kfk[Φ], 105 CW0,q , 312 kfk(Φ), 105 CY , central Young, 51 kµk[p], 1 ≤ p < ∞, 118 Cb, continuous, bounded, 398 kµk(p), 1 ≤ p < ∞, 14, 118 c , 22 ← 0 kµk[p], 1 ≤ p < ∞, 437 ← kµk[p], 1 ≤ p < ∞, 437 D, 23 670 Index of Notation

loc DWp, 142 Hα , α-Hölder locally, 274 Dλ, 348, 646 Hα,∞, space of bounded α-Hölder DkF , kth derivative of F , 252 functions, 219, 274 0 Dm, Dirichlet kernel, 555 Hφ, 159 ∗ Dm, simplified Dirichlet kernel, 555 HK, Henstock–Kurzweil, 57 D(ν)F , derivative in measure, 358 dkF , kth differential of F , 252 I, class of interval functions, 19 + ∆[a,b]f, 17 Ip, of bounded p-variation, 13 − ← ∆ f, 17 Ip , 436 [a,b] ← ± Ip = Ip ∩ Ip , 437 ∆[a,b]f, 17 IΦ, of bounded Φ-variation, 117 ∆+f, right jump, 17 IΦ, vector space, 118 ∆−f, left jump, 17 e ± IP (J), interval partitions of J, 12 ∆ f, two-sided jump, 17 I(J), subintervals of J, 11 Ios, open and singletons, 11 F, filtration, 573 F(X), filtration generated by X, 573 ← K, field R or C, 9 F , quantile function, 616, 647 κ, point partition, 12 Fn, empirical distribution function, 616 (a) |κ|, mesh of partition κ, 12 f− , 51 (a,b) p f− , 52 L (Ω; X), 1 ≤ p < ∞, 14 (a,b) ∞ f+ , 52 L (Ω; X), 14 (b) LΦ(Ω; X), 15 f+ , 51 ≻ L(ν) =ν ˇ, 438 {fn }, forward Picard iterates, 510 ≺ ℓ∞, universally measurable, 398 {fn }, backward Picard iterates, 511 U p f(t+), right limit, 17 ℓ , 1 ≤ p < ∞, 14, 191 ∞ f(t−), left limit, 17 ℓ , 10 ℓΦ, 15, 563

G∞, bounded locally, 338 L, bounded linear operators, 10 Gβ, β growth condition, 338 LS, Lebesgue–Stieltjes, 4 ≻ Lk(X,Y ) = L(X,Lk−1(X,Y )), 252 {gn }, forward Picard iterates, 510 ≺ L(kX,Y ) = M (X,...,X; Y ), 240 {gn }, backward Picard iterates, 511 k p Γ∞, essential supremum locally, 338 L (Ω; X), 1 ≤ p < ∞, 14 k Γβ, β growth condition a.e., 338 Ls( X,Y ), symmetric elements, 240 L(s), symmetrization of L, 241 ∞ H, class of stochastic processes, 629 L (Ω; X), 14 0 HWGα,q = UHα ∩ WGα,q, 508 L (Ω; X), 15 Φ HW n+α,q, 313 L (Ω; X), 15 HWloc , 313 Lµ,a, 20 n+α,q Λ(φ, ψ; η), 157 HW 0,α,q, 312 loc HW 0,α,q, 313 MIp, 476 HW α,q, 305 M , bounded k-linear maps, 239 loc k HW α,q , 307 M(β, γ), 603 Hn+α, nth derivative α-H¨older, 375 M k[h], k-linear multiplication operator H1+α, 275 induced by h, 245, 275 loc H1+α, 275 Mp, 486 ← H1+α,∞, 404 Mp , 493 Hα, set of α-H¨older functions, 11 µh = µh,[[a,b]], 19 Index of Notation 671

µ, strict product integral, 433 TC, two-function composition, 393 b ≻ µˇ, additive transform, 438 Tc,u, forward integral transform, 509 ≻ µ∗ν, convolution, 196 Tc,u, backward, 510 |τ|, mesh of tagged partition τ, 12 NF , autonomous Nemytskii operator, Θ(φ, ψ), 171 273 Nψ, Nemytskii operator, 273 UCR, 508 UGβ, s-uniform β growth, 508 OscA(f), oscillation of f over A, 10 UHα, s-uniformly α-Hölder, 301 Osc(µ; J) oscillation of µ over J, 23 loc UHα , locally, 301 Umf, partial Fourier sum, 555 P, product integral operator, 457 υ(f;[a, b]), index of p-variation, 588 P(µ) = (1I + dµ), 438 P·,·(·, ·), Markov kernel, 601 V, 103 PP (S), point partitions of S, 12 VM, 116 P k, k-homogeneous polynomials, 242 Vp(f), 13 P (µ; A, A), product, 408 t Vp(µ), 1 ≤ p < ∞, 452 a(I + dµ), product integral, 408 VΦ(µ), 452 t v (f), p-variation of f, 13 a(I + dh), product integral, 9 p v¯p(f), 147 ≻ Qc,u, forward integral transform, 509 vp(µ), p-variation of µ, 13, 117 ≺ ← Qc,u, backward, 510 vp (α; A), 436 ∗ ∗ vp (f), p -variation of f, 142 R, class of regulated functions, 17 ∗ ∗ vp (µ), p -variation of µ, 121 RRS, refinement Riemann–Stieltjes, 24 vΦ(X) = {vΦ(X; t)}t≥0, 573 RS, Riemann–Stieltjes, 24 vΦ(f), Φ-variation of f, 103 RYS, refinement Young–Stieltjes, 29 vΦ(µ), Φ-variation of µ, 117 Rem, remainder, 237

Rµ,a, 20 WGα,q, 508 r(·), spectral radius, 219 Wp, bounded p-variation, 13 ∗ ∗ ran , range, 77 Wp , having p -variation, 142 W∞ = R, regulated, 183 ≻ S[a,·), solution mapping, 483, 537 Wα,q, 305 S ≻ , solution mapping, 483, 536 loc (a,·] Wα,q , 307 ≺ S(·,b], solution mapping, 483, 537 Wβ,p, 197 ≺ S[·,b), solution mapping, 483, 537 WΦ, bounded Φ-variation, 103 (S), full Stieltjes, 47 WΦ, Banach space, 105 floc SK , Kolmogorov sum, 32 WΦ , 573 SLC , left Cauchy sum, 632 WPS, Ward–Perron–Stieltjes, 63 SMS , mean Stieltjes sum, 631 Wp(µ), 485 ← SRS , Riemann–Stieltjes sum, 24 Wp (µ), 493 SYS , Young–Stieltjes sum, 29 wB (f), semivariation of f, 26 sp(f; κ), p-variation sum, 13 wB (µ), semivariation of µ, 41 sp(µ; A), p-variation sum, 13 w⋆(µ), 43 ← sp (α; A), 436 sΦ(f; κ), Φ-variation sum, 103 X+ = {X+(t), t ≥ 0}, 572 sΦ(µ; A), Φ-variation sum, 117 X− = {X−(t), t ≥ 0}, 572 σp(µ), 121 ∗ σp (µ), 121 YP , Young interval partitions, 121