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16 MA THEMA TICS: S. KAKUTA NI PROC. N. A. S. REPRESENTA TION OF MEASURABLE FLOWS IN EUCLIDEAN 3- SPA CE By SHIzuo KAKUTANI

INSTITUTE FOR ADVANCED STUDY Communicated November 19, 1941

1. The purpose of this note is to show that every measurable flow defined on a measure space satisfying certain separability conditions is isomorphic to a continuous flow on a subset of a Euclidean 3-space with ordinary Lebesgue measure. 2. We begin with definitions.1 A measure space 0(t, m) is a system of a space 2, a Borel field e of subsets M of Q, and a countably additive completed measure m(M) defined on e3 such that 0 < m(Q) < co. A set belong- ing to e3 is called measurable. A measure space Q(j9, m) is properly separable if there exists a countable collection 2l of measurable sets, called a basis, such that, for any M e 5Q and for any e > 0, there exists a subsequence {M1 (n = 1, 2, . from such that M C z Mn and z m(M") < m(M) + I n = 1 n = 1 e. A countable collection 2l of subsets of Q is a separating sequence of Q if, for any two distinct points of Q2, there exists at least one set from 2I which contains one but not both of them. If Q is a Lebesgue measurable subset of a Euclidean n-space Rn with a finite Lebesgue measure, and if V(Q) is the collection of all Lebesgue measurable subsets of Q, then Q(V(i), ,I) (,I means n-dimensional Lebesgue measure in R') is called a Lebesgue measure space. More generally, if Q is an arbitrary subset of a Euclidean n-space Rn with a finite Lebesgue outer measure, and if 2*(Q) is the collection of all subsets M of Q1 of the form: M = £l.A, where A is a Lebesgue measurable set of R', then Q(2*(&l), ,*) (A* means n-dimensional Lebesgue outer measure in R') is called a Lebesgue* measure space.2 Clearly, Lebesgue and Lebesgue* measure spaces are properly separable and have a separating sequence of measurable sets. A measure preserving transformation (of a measure space onto another or onto the same measure space), a flow (= one parameter group of measure preserving transformations of a measure space onto itself), and the notions of invariant sets and ergodicity are defined as usual. A flow I T I (- < t < + co) defined on a measure space Q(9, m) is measurable if, for any M e 9, the set M = [(w, t): T,(w) e M] is measurable in the product space of Q with the real t-axis, where measure is defined multiplicatively in terms of the given measure m(M) on Q and the Lebesgue measure on t-axis. Two flows SI} and ITJ defined on two measure spaces Q(9, m) and Q'(QY, m'), respectively, are isomorphic to each other if it is possible to find two in- Downloaded by guest on September 28, 2021 VOL. 28, 1942 MA THEMA TICS: S. KAK UTANI 17 variant sets of measure zero N e Q3, N' e Z' and a measure preserving transformation of -N onto ' - N' which carries I SI} into IT1. 3. Our main results are: THEOREM 1. Let ITj} be a measurable flow defined on a properly separable measure space having a separating sequence of measurable sets. If every point of the space is of measure zero, then I T7 I is isomorphic to a continuous flow on a Lebesgue* measure space in a Euclidean 3-space R.' THEOREM 2. Every measurableflow defined on a Lebesgue measure space in R' (n > I) is isomorphic to a continuous flow on a Lebesgue measure space in R3. It is desirable to represent IT,} as a differentiable or an analytic flow in R3. But this is not always possible. For instance, if I Tt} is a group of rotations of a circumference (with one-dimensional Lebesgue measure), then the corresponding representation space in R3 must be a simple Jordan curve with a positive three-dimensional Lebesgue measure. It is, however, easy to see that such a curve cannot be differentiable. The proof of Theorem 1 will be given in sections 5, 6 and 7, and Theorem 2 will be proved in section 6. 4. Let 9(V, m) be a measure space, and let S be a measure preserving transformation of Q(Q8, m) onto itself. Let furtherf(c) be a positive valued measurable and integrable function defined on Q. We assume that there exists a constant c > 0 such that f(c) 2 c for all w e Q. Consider the product space of Q with the real u-axis, with the measure m(M) defined on it multiplicatively in terms of the measure m(M) on Q and the Lebesgue measure on the u-axis. Let Q be the portion of this product space under the graph of f(w), i.e. the set of all points of the form co = (w, u), w e Q, 0 < u < f(c), and let e3 be the collection of all m-measurable subsets M of Q. Then QM, m) is a measure space and the flow I Tt } defined on it by3 Tt(co, u) = (co, u+ t), if-u t < -u +f(w), = (S(c), u+ t -f() - -f(S. -1..))) n - 1 n if -u + z f(S&(w)) < t < -u + lf(Sk(w)), n = 1, 2,. k = O k = O = (S (W), u+ t +f(S1 (co)) + f.s..M)),+ if -U - f(S-k(Co)) < t < -u - 2 f(S-k(W)), n = 1, 2, k=1 k=1 is called a flow built under a function. f(w) is called the ceiling function, Q2 and S are called the base space and the base transformation, respectively. 5. It was proved by W. Ambrose4 that every ergodic measurable flow (defined on an arbitrary measure space) is isomorphic to a flow built under a function. Later it was shown5 that the same thing is true for general non- ergotic measurable flows (defined on a measure space having a separating sequence of measurable sets) if we permit that the base space has an infinite Downloaded by guest on September 28, 2021 18 MA THEMA TICS: S. KAK UTA NI PROC. N. A. S. measure. More precisely speaking, given a measurable flow I T1 I defined on a measure space 9(5, m) having a separating sequence of measurable sets, it is possible to decompose Q into a countable number ofdisjoint invariant measurable sets: Q = 2, (some of the terms may be missing) in such a way that (i) { T1 I is the identity on Qo, and (ii) I T1 I is isomorphic to a flowvbuilt under afunction on each Qn (n = 1, 2, . .. ). Thus the proof of Theorem 1 is reduced to the discussion of the special cases (i) and (ii). Since the first case can be easily discussed,6 we shall dis- cuss only the second case. Hence, from now on, we assume that I T1 } is a flow built under a function, and we adopt the notation of section 4. 6. By assumption, there exists a constant c > 0 such that f(w) _ c for all w E Q. We may also assume thatf(w) . 2c for all co e Q. Next we remark that the base space Q(, m) is properly separable and has a separating sequence of measurable sets.7 Let a = {M," ( n = 1, 2, . . . ) be a basis of Q(3, m) which is at the same time a separating sequence. We may assume that a contains all sets M of the form: M = [w: a < f(w) < b], where a and b are arbitrary rational numbers. Further we may assume that a is invariant under the base transformation S (i.e., M e a implies S(M) e a and S-1(M) e ). Finally, we may assume that a is a finite field. Let us now define the distance of two points w and co' of Q by d(w, co') = OD 2 3-f (Pn ) p-n(w') |, where Vn(w) is the characteristic function of M,. It is easy to see that, with respect to this metric, Q is a zero-dimensional separable metric space and that the ceiling function f(w) is a continuous function. Moreover, w' = S(w) is a homeomorphism of Q onto itself, and the following conditions are satisfied: (1) every open set (and hence every Borel set) is measurable, (2) every measurable set is contained in a Borel set of the same measure. Let us consider the set Qi° of all points X of the form: X = (w, u), co eQ O < u _ f(w), and define the distance of two points X = (w, u) and ' = (W', u') of Q° by d(c, c'') = d(co, w') + u - u'|. Then Q° is clearly a one- dimensional separable metric space. If we identify two points (co, f(w)) and (S(w), 0) of Q° for each w e Q2, then we have a new metric space. This may be observed as a metrization of Q, and it is not difficult to see that, with respect to this metrization, Q is a one-dimensional separable metric space, on which IT, I is a continuous flow. It is also not difficult to see that the conditions (1) and (2) are satisfied by the measure space Q(, m). Moreover, by assumption, (3) every point is of measure zero; but it is not necessarily true that (4) every open set is of positive measure. Downloaded by guest on September 28, 2021 VOL. 28, 1942 MA THEMA TICS: S. KAK UTA NI 19 Let N be the union of all open sets of zero measure of Q, if there are any. Then N is an invariant null set of measure zero, and the remaining part - N clearly satisfies the conditions (1), (2), (3) and (4). Hence we may assume that Q itself satisfies all these conditions. 7. By a well-known theorem,8 we can embed Q homeomorphically into a bounded part of a Euclidean 3-space R3, and thus we obtain a representation of the given flow as a continuous flow defined on a subset of R3. The proof of our theorem, however, is not yet finished since the measure space Q(Q', m), thus embedded into RI, does not necessarily coincide with the Lebesgue* measure space Q(C*(j), ,*). Let Q* be a bounded Borel set of R3 such that Q C Q* and ,u*(Q) = ,u(Q*), where ,u and ,u* mean Lebesgue and Lebesgue outer measures in R3. Let Q be a closed cube in R3 which contains Q* entirely in its interior, and let d* be the collection of all subsets M of Q such that M * Q e e and M -MM * is Lebesgue measurable. Z3* is clearly a Borel field, and if we put v(M) = m(M.i) + A(M -MM Q*), then v(M) is a countably additive completed measure defined on Q6*. It is clear that the condition (1) is satisfied by the measure space Q(Q3*, v), but the condition (2) is not necessarily satisfied. Let Z** be the collection of all subsets A of Q such that there exist two Borel sets A1 and A2 with A1 C A C A2 C Q and v(Al) = v(A) = v(A2). Then Q** is a Borel subfield of $&, and the measure space Q(Z**, v) clearly satisfies the conditions (1) and (2). Since the conditions-(3) -and -(4) are clearly satisfied by the measure space Q(3**, v), and since the boundary of Q is a set of v-measure zero, there exists, by a theorem of J. C. Oxtoby-S. M. Ulam9 and J. von Neumann, a homeomorphism c' = h(c) of Q onto another closed cube R, which carries the measure space Q(Q**, v) into the Lebesgue measure space R(V(R), A). Since Q(513*, v) is an extension of Q(3**, v), it will be carried over by this homeomorphism w' = h(w) into a measure space which is an extension of the Lebesgue measure space R(V(R), A). We denote this measure space by R(V(R), ji), where A is an extension of the ordinary Lebesgue measure defined on a Borel field V(R) containing all Lebesgue measurable subsets of R. Let Q' = h(Q) be the image of Q. by h(w), and let us consider the part of R(V (R), ,.) restricted on Q', which we shall denote by IA'(V('), ,u). Since it is clear that the measure space -'(A'(Q'),,) is isomorphic to the given measure space Q(M, mi), the proof of our theorem will be finished if we can prove that Q'(e(Q'), 8 is the same measure space as the Lebesgue* measure space

For this purpose, let M' E S*(Q'). Then, by definition, there exists a Lebesgue measurable set A' such that M' = A' - '. We may assume that A' C Q*' = h(Q*), and ,*(M') = ,u(A'). Let A' and A' be two Borel sets such that A' C A' C A' C Q*' and M(Al) = ,(A') = M(A'). Let further M, A1, A and A2 be the inverse images of M', A, A' and A4 by h(w). A1 and A2 Downloaded by guest on September 28, 2021 20 MA THEMA TICS: S. KAK UTANI PROC. N. A. S. are Borel sets, and we have v(Al - A2) = 0 since O(Al - A) = 0. Since A1 C A C A2, we have A e t}**, and consequently M = A-Q e 8*, which in turn implies that M' = h(M) e (§'). Thus we have proved that V*(Q') C (Q '). Moreover, we have ,u*(M') = u(A') = v(A) _ v(M) = m(M) = A(M). Conversely, let M' e(Q'). Then there exists an M C Q such that M' = h(M) and M e VI*. Since M C Q, we must have Me Z. Hence there exist two Borel sets A1andA2such that A1 -C M CA2 *Q and m(Al Q) = mr(M) = mr(A2- Q). We may assume that A1 C A2cQ*C. Let A be an arbitrary set such that A1 C A C A2 and M = A * Q. Let further A', A' and A' be the 'mages of A1, A and A2 by h(w). Then A' and A; are Borel sets, and, since (A2- A1) = mr((A2 - A1)Q) = O, we have A(A2-A) = O. Since A' C A' C A', we have A' e ?(Q), which together with the relation M' = A' *Q2' will imply M' e£( '). Thus we have proved that w(') C £*(Q'). More- over, we have 7(M') = m(M) = v(A) = ,i(A') _ 1I*(M'). Combined with the inequality obtained above, this will give ,(M') = .j*(M') for all M' e ?(Q') = 3I(Q'), and the proof of Q'(3(Q%'), A) = '(2*(Q%'), ,u*) is completed. 8. In order to prove Theorem 2, we need only prove the following10: LEMMA. Ifa LebesgueI* measure space UI( *(Q*), ,I*) in Rm(m > 1) is isomorphic to a Lebesgue measure space I(A3(i),,u) in R' (n _ 1), then Q* is a Lebesgue measurable subset of Rm. Proof. Let w* = p(w) be a measure preserving transformation of Q(2(2), ,u) onto I'(2I(i*),,I') which gives this isomorphism. p(cW) is clearly Lebesgue measurable. Hence there exists a Borel measurable mapping 46(w) such that sp(w) = 4'(w) almost everywhere on Q. Let go be a Borel set such that go C Q, ,u(Q -Qo) = 0 and o(cw) = ),6(w) everywhere on go. Then Q* = (,o() = (p(9o) + sP(Q -o) = ,t(Q20) + p(Q - no). Since 4P(s2o) is Lebesgue measurable as the image of a Borel set go by a Borel measurable mapping ,(4w), and since so(Q -Qo) is of Lebesgue measure zero as the image of a set of measure zero -QO by a measure preserving transformation sp(w), Q* must be measurable.

1 For details of definition, see Ambrose, W., and Kakutani, S., "Structure and Con- tinuity of Measurable Flows," to appear in Duke Math. Jour. The author is much in- debted to Dr. W. Ambrose for his kind conversations on the subjects discussed in the present paper. 2 See Doob, J. L., "Stochastic Process Depending on a Continuous Parameter," Trans. Amer. Math. Soc., 42, 107-140 (1937). 8 See Ambrose, W., "Representation of Ergodic Flows," Ann. Math., 42, 723-739 (1941). 4 See reference 3 above. 6 See reference 1 above. 6 In this case, we have only to prove that every properly separable measure space, which has a separating sequence of measurable sets and on which every point is of mea- Downloaded by guest on September 28, 2021 VOL. 28, 1942 MATHEMATICS: R. P. BOAS, JR. 21

sure zero, is isomorphic to a Lesbesgue* measure space in R3. We can even prove that such a measure space is isomorphic to a Lebesgue* measure space in R1. 7 See reference 1 above, Lemmas 5 and 6. 8 This embedding can be carried out in a concrete way. First we map the base space Ql onto a zero-dimensional set X in the interval 0 _ x _ 1 by x = 2 3-" 4,(w), where '4,(W) is the characteristic function of M". Then fQ° can be considered as a set in the (x, u)- plane. The final embedding of a into RI can be obtained by twisting Q° in R3 and by identifying the points which correspond to (w,f(w)) and (S(w), 0). 9 Oxtoby, J. C., and Ulam, S. M., "Measure Preserving Homeomorphism and Metrical Transitivity," Ann. Math., 42, 874-920 (1941). Theorem 21. 10 Theorem 2 is also true if the given measure space Q(1, m) is normal in the sense of Halmos, P., and von Neumann, J., Bull. Amer. Math. Soc., 47, 696-697 (1941).

INVERSION OF A GENERALIZED LAPLACE IN1EGRAL BY R. P. BOAS, JR. DEPARTMENT OF MATHEMATICS, DuKB UNIVERSITY Communicated December 5, 1941 The transform f (z) (2 )f t1/2K,(zt)da(t), (1) in which K,(x) denotes a Bessel function of imaginary argument,' has recently been discussed by C. S. Meijer2 and R. E. Greenwood.' This transform is a generalization of the Laplace transform, to which it reduces when v = '1/2-. Meijer obtained an inversion formula for (1) generalizing the complex inversion formula for Laplace transforms. The object of this note is to point out that (1) can also be inverted by means of a differential operator which is a generalization of the Post-Widder inversion operator4 for Laplace transforms. In addition, necessary and sufficient conditions for the representation of a given function f(z) in the form (1) can be ex- pressed in terms of the inversion operator.5 Let us define differential operators W, Qk, by the relations W[g(z)] = v- 1[Zl- 2'g,(Z)] ', 1 - Qk[f (u)] = (2k)k2k + 3/2 Wk[Z - 1/2f(Z)] IZ = 2k/u We suppose that -1/2 < (P) < 1/2, and that (1) converges for some z (and hence for any z' with larger real part). THEOREM 1. If a(t) is the indefinite integral of p(u), then for almost all positive u so(u) = lim Qk[f(u)]. (2) k - Go Downloaded by guest on September 28, 2021