A Note on the Σ-Algebra of Cylinder Sets and All That

A note on the σ-algebra of cylinder sets and all that

Jose´ Luis Silva CCM, Univ. da Madeira, P-9000 Funchal Madeira BiBoS, Univ. of Bielefeld, Germany ([email protected])

September 1999 Abstract

In this note we introduced and describe the different kinds of σ-algebras in inﬁnite dimensional spaces. We emphasize the case of Hilbert spaces and nuclear spaces. We prove that the σ-algebra generated by cylinder sets and the Borel σ-salgebra coincides. Contents

1 1 Introduction

In many situations of infinite dimensions appear measures defined on different kind of σ-algebras. The most common one being the Borel σ-algebra and the σ-algebra generated by cylinder sets. Also many times appear the σ-algebra associated to the weak topology and as well the strong topology. Here we will be concern mostly with the first two, i.e., the σ-algebra of cylinder sets and the Borel σ-algebra. In applications we find also not unique situations where one should have the above mentioned σ- algebras. Therefore we will treat the most common described in the literature. Namely the case of Hilbert riggings and nuclear triples. Rigged Hilbert spaces is an abstract version of the theory of distributions or generalized functions, therefore this case is very important it self. As it is well known nuclear triples appear often in mathematical physics, hence we will also consider it. Booths cases are well known in the literature but for the convenience of the reader and completeness of this note we include on the appendix its definitions. Neverthe- less we recommended the interested reader to look the classical book of (?) for more details and historical remarks. Thus in Section 2 we describe the σ-algebra generated by cylinder sets on rigged Hilbert spaces. Moreover we prove that, in fact, that this σ-algebra coincides with the Borel σ-algebra. In Section 3 we obtain analogous results for nuclear triples.

2 Case of rigged Hilbert spaces

Let ( , (., .)) be a Hilbert space and consider ( ( , )) another Hilbert space which H H+ · · is such that densely. Then we can consider the triple H+ ⊂ H

+ (1) H− ⊃ H ⊃ H which is called rigged Hilbert space, see Appendix A.1 for more details or (?), (?), (?), and (?). Now we would like to introduce a σ-algebra in . To this end we proceed as H− follows: consider the following collection of ﬁnite dimensional subsets from H = L L , dim(L) < (2) L { | ⊂ H ∞} This family is such that the following conditions holds:

1. if L , then an arbitrary subspace L˜ L belongs to , ∈ L ⊂ L 2. if L , L , l.h. L , L , where l.h. L , L = αl + βl α, β 1 2 ∈ L { 1 2} ∈ L { 1 2} { 1 2| ∈ R, l1 L1, l2 L2 , ∈ ∈ }

3. L L is dense in . ∈L H ! 2 For each L and any A (L) (the Borel σ-algebra on L) we deﬁne a cylinder ∈ L ∈ B set with coordinate L and base A by

C (L, A) := ξ PL (ξ) A . (3) { ∈ H−| ∈ } Here PL is the projection on which can be extended to . More precisely, let H H− ej, j N be an orthonormal bases in + which is also orthonormal in . Then any { ∈ } H H element h has the following decomposition: ∈ H ∞ h = (h, ej) ej, j=1 " this implies with dim(L) = N

PL (h) = (h, ej) ej. j=1 " On the other hand, any ξ is the limit in the norm of a Cauchy sequence of ∈ H− | · |− elements in i.e., (hn)∞ such that ξ hn 0, n . Therefore any H ∃ n=1 ⊂ H | − |− → → ∞ element ξ also admits a decomposition in the basis (ej)∞ as ∈ H− j=1 ∞ ξ = ξ, e e . (4) ( j) j j=1 " Hence we easily ﬁnd that

N P (ξ) = ξ, e e , N = dim (L) . (5) L ( j) j j=1 " This implies that, indeed, the cylinder set (??) is well defined. We denote by (L) the C σ-algebra of cylinder sets with a fixed coordinate L, i.e., (L) = (L, A) A (L) . C {C | ∈ B } Then we can define the algebra of cylinder sets by

( ) = (L) . CL H− C L #∈L And ﬁnally the σ-algebra of cylinder sets is given by

σ ( ) = σ ( ( )) , C H− CL H− that means, the σ-algebra generated by the algebra of cylinder sets. We would like to stress that in fact ( ) is only an algebra and not a σ-algebra. This can be seen with C H− the following example: consider the following family of cylinder sets C(L , A ), i = 1, 2, . . . , { i i } 3 where each L is ﬁnite dimensional and A (L ). Then the set i i ∈ B i

∞ K = C (Li, Ai) i=1 $ is not a cylinder set because the coordinate is not ﬁnite dimensional.

Proposition 2.1 For any class we have σ ( ) = ( ), i.e., σ ( ) is inde- L C H− B H− C H− pendent of . L

Proof. First we prove that σ ( ) ( ): in fact to see that every cylinder C H− ⊂ B H− set belongs to ( ) we take into account the following representation for C (L, A) B H− which is a consequence of (??):

C (L, A) = ξ ( ξ, e1 , ..., ξ, en ) A . { ∈ H−| ( ) ( ) ∈ } To prove the other inclusion we proceed as follows: let e−, j N be an orthonormal { j ∈ } basis in . Then consider the following sets in : H− H− N ∞ B¯− (0, R) = ξ (ξ, en) R , R > 0 ∈ H−| − ≤ N=1 % n=1 & $ " which is the closed ball in with radius R. The open ball is obtained as H− ∞ 1 B− (0, R) = B¯−(0, R ) − n n=1 # which belongs to σ ( ) because each B− (0, R) is an element of σ ( ) . Since C H− C H− any open set in is the union of open balls, then it belongs to σ ( ). This implies H− C H− that ( ) σ ( ). ! B H− ⊂ C H−

3 Case of nuclear triples

The case of nuclear triples is the most common in applications, e.g., the Schwartz triple, tempered distributions, etc. Hence we suppose given a nuclear triple

$ , (6) N ⊃ H ⊃ N where is a Hilbert space, is a dense subset in which is nuclear, see Appendix ?? H N H for a detailed description of (??) or (?), (?), and (?). Hence we will take a family of subsets from each being of ﬁnite dimension: N = L dim (L) < . (7) L { ⊂ N | ∞}

4 Given L we deﬁne a cylinder set with coordinate L and base A (L) by ∈ L ∈ B C (L, A) = Φ $ P (Φ) A . (8) { ∈ N | L ∈ } As before P is an orthogonal projection onto L deﬁned in which extends to $ L H N by continuity as in the previous case. Sometimes it is useful to have also cylinder sets n of the form (??) in coordinates. Thus we choose a basis (ej)j=1 in L which is also orthogonal in . Then for any Φ $ we have N ∈ N n P (Φ) = Φ, e e . L ( j) j j=1 " This implies the following representations for C (L, A):

C (L, A) = Φ N $ ( Φ, e , ..., Φ, e ) A . { ∈ | ( 1) ( n) ∈ } We proceed introducing the σ-algebra of cylinder sets with a ﬁxed coordinate L ∈ by L (L) = C (L, A) , A (L) . C { ∈ B } Then the algebra of cylinder sets in $ we denote by ( $) and is deﬁned by N C N ( $) := (L) . C N C L #∈L Finally the σ-algebra of cylinder sets is the σ-algebra generated by (N $) , i.e., C ( $) = σ ( ( $)) . Cσ N C N Proposition 3.1 If the space is a countable Hilbert space, then we have N ( $) = ( $) = ( ) , Cσ N Bw N Bs N where ( $) (resp. ( )) is the Borel σ-algebra degenerated by the weak (resp. strong) Bw N Bs N topology. Proof. The prove of these facts can be founded in (?, Appendix 5) or (?). !

A Rigged Hilbert spaces

The concept of rigged Hilbert spaces was introduced as an abstraction of the theory of generalized functions of the type of Sobolev-Schwartz, i.e., d 2 d d $(R ) L (R ) (R ), D ⊃ ⊃ D or d 2 d d S$(R ) L (R ) S(R ). ⊃ ⊃ Let ( , ( , ) ) be a complex Hilbert space and assume additionally that ( , ( , ) ) H0 · · 0 H+ · · + is another complex Hilbert space such that % is dense and H+ → H0 ϕ c ϕ , ϕ , c > 0. (A.1) | |0 ≤ | |+ ∀ ∈ H+

5 Remark A.1 The inequality (??) may be reduced to the following inequality

ϕ ϕ $ , ϕ , (A.2) | |0 ≤ | |+ ∀ ∈ H+ where ϕ $ is an equivalent norm in . Hence, in this appendix we suppose without | |+ H+ lost of generality that the inequality (??) veriﬁes with c = 1.

Remark A.2 The elements from 0 will be denoted by f, g, h, and we call them H · · · ordinary functions, while the elements from play the role of test function. We H+ denote the elements from by ϕ, ψ, η, . . .. H+ Let f be a ﬁxed element. Deﬁne a functional l associated to f on by ∈ H0 f H+ lf : + : C, ϕ lf (ϕ) := (ϕ, f)0. (A.3) H → ,→

Proposition A.3 The functional lf has the following properties.

1. Is linear, i.e., α, β C, lf (αϕ + βψ) = αlf (ϕ) + βlf (ψ). ∀ ∈ 2. Is continuous, i.e., l (ϕ) f ϕ . | f | ≤ | |0| |+ ¯ 3. α, β C, f, g 0, lαf+βg(ϕ) = α¯lf + βlg. ∀ ∈ ∀ ∈ H Proof. These properties follows immediately from the definition (??). ! Let us define the following mapping in H0 : 0 R+, f f := lf := sup lf (ϕ) , ϕ +, ϕ + = 1 . (A.4) | · |− H → ,→ | |− - - {| | ∈ H | | } Proposition A.4 The mapping verifies the properties of a norm. | · |−

1. f 0, f 0. | |− ≥ ∀ ∈ H 2. λf = λ f , λ C, f 0. | |− | || |− ∀ ∈ ∀ ∈ H

3. f + g f + g , f, g 0. | |− ≤ | |− | |− ∀ ∈ H 4. In f = 0, then f = 0. | |− Proof. Again the proof of the above proposition follows by applying the deﬁnition of . ! | · |−

Deﬁnition A.5 We deﬁne as the completion of 0 with respect to , i.e., := H− H |·|− H− 0|·|− . The space plays the role of generalized functions and its elements will be H H− denoted by Φ, Ψ, Θ, . · · · Remark A.6 Consider the sesquilinear form

, : + + C, (ϕ, ψ) ϕ, ψ := (ϕ, ψ)0. (· ·) H × H → ,→ ( )

6 1. Then , can be extended to + , i.e., we can deﬁne ϕ, Φ := (ϕ, Φ)0, (· ·) H × H− ( ) ϕ +, Φ . Note that ϕ, Φ is well deﬁned and independent of the ∀ ∈ H ∀ ∈ H− ( ) sequence (f )∞ which converges to Φ. n n=1 ⊂ H 2. The following generalization of the Cauchy-Schwarz inequality is valid

(ϕ, Φ)0 ϕ + Φ . | | ≤ | | | |H− Theorem A.7 The space is a Hilbert space. H− Proof. It is enough to show that the norm is given by a scalar product, i.e., | · |− f 0, we have f = (f, f) . Denote by O the embedding operator from + ∀ ∈ H | |− − H into and its dual by I, i.e., H '

O : % , ϕ Oϕ := ϕ, I O∗ : , f If, H+ → H0 ,→ ≡ H0 → H+ ,→ such that (f, ϕ) = (f, Oϕ) = (If, ϕ) , f , ϕ . 0 0 + ∈ H0 ∈ H+ Then, we have

f = sup (f, ϕ)0 , ϕ +, ϕ + = 1 | |− {| | ∈ H | | } = sup (If, ϕ) , ϕ , ϕ = 1 {| +| ∈ H+ | |+ } = If . | |+

Hence, if we deﬁne the scalar product ( , ) in 0 by · · − H

(f, g) := (If, Ig)+, f, g 0, − ∈ H 2 we obtain f = (f, f) . Therefore the pre-Hilbert space ( 0, ( , ) ) after comple- | |− − H · · − tion turns into a Hilbert space which coincides with . ! H− Remark A.8 1. The scalar product ( , ) can be written as · · −

(f, g) = (If, g)0, = (f, Ig), f, g 0. − ∈ H 2. Since O = I = 1, then - - - -

f f 0, f 0. | |− ≤ | | ∈ H 3. The scalar product in is given by H−

(Φ, Ψ) := lim (fn, gn) , H− n − →∞

where (fn)n∞=0, (gn)n∞=0 0 are Cauchy sequences in the norm converging ⊂ H |·|− to Φ and Ψ, respectively.

7 4. Since f, g 0 (f, g) = (f, g) = (If, Ig)+, then I : D(I) = 0 ∀ ∈ H H− − H ⊂ + is an isomorphism between and + deﬁned in a dense set. We H− → H H− H denote its extension by continuity by I˜. Hence I˜ is a unitary operator from H− onto . H+ 5. The operator I˜ is continuous.

6. We have the following equality

(ϕ, Φ)0 = (ϕ, I˜Φ)+, ϕ +, Φ . ∀ ∈ H ∀ ∈ H−

Theorem A.9 1. The space coincides with the dual of +, with respect to 0, H− H H i.e., = ( +)$. (A.5) H− H

2. The dual of coincides with + with respect to 0. H− H H Remark A.10 We have constructed the following chain

0 + (A.6) H− ⊃ H ⊃ H called Rigged Hilbert spaces or Gelfand triple.

1. The fact that = ( +)$ means that l ( +)$ Φl such that H− H ∀ ∈ H ∃ ∈ H− l(ϕ) = (ϕ, Φ ) , ϕ , l 0 ∀ ∈ H+

and we say that is the dual of + in terms of 0 which is reﬂected in the H− H H representation of the functional l ( )$ using ( , ) . ∈ H+ · · 0

2. The classical identiﬁcation = ( )$ says that the functional l ( )$ has H+ H+ ∀ ∈ H+ representation in terms of the scalar product in , i.e., H+ l(ϕ) = (ϕ, ψ ) , ϕ, ψ . l + l ∈ H+

Deﬁnition A.11 If O : + 0 is an Hilbert-Schmidt operator (or quasi nuclear), H → H then the triple (??) is called nuclear.

2 2 Example A.12 Let 0 = L (µ) := L (X, (X), µ) with µ(X) < and + = H B ∞ H L2(pµ) with p : X [0, ] a measurable ﬁnite µ-a.e. Then the corresponding triple → ∞ is 2 1 2 2 L (p− µ) L (µ) L (pµ). (A.7) ⊃ ⊃

8 B Nuclear triples

Nuclear spaces are essentially inﬁnite dimensional. In inﬁnite dimensions closed ball is not pre-compact (or in other words, from a sequence we can not take a convergent subsequence). In the case of nuclear spaces we have the following very important lemma.

Lemma B.1 Let be a nuclear space. Then any bounded set A is pre-compact, N ⊂ N i.e., after closure it is compact.

Proof. For the proof see e.g., (?, pag. 55). ! Let T be an arbitrary set of indices and ( τ , ( , )τ )τ T a family of Hilbert spaces H · · ∈ indexed in T . We assume that the family of Hilbert spaces satisﬁes the following conditions:

(DE) The family ( τ , ( , )τ )τ T is directed by embedding, i.e., τ, τ $ T , τ $$ T H · · ∈ ∀ ∈ ∃ ∈ such that the embedding

i : % , i : % τ "",τ Hτ "" → Hτ τ "",τ " Hτ "" → Hτ " are continuous.

(D) The linear space given by N := N Hτ τ T $∈ is dense in each , τ T . Hτ ∈ In we introduce the projective limit topology, i.e., the weakest topology in N N such that the embedding i : % , τ T are continuous. As a system of base τ N → Hτ ∈ neighborhoods we can take the following collection

Σ = U(ϕ, , ε) ϕ , τ T, ε > 0 , (B.1) { | ∈ N ∈ } where U(ϕ, , ε) = ψ ϕ ψ < ε , { ∈ N | | − |τ } i.e., the open ball in are intersections of with open balls in each . N N Hτ

Deﬁnition B.2 1. The space = τ T τ with the topology generated by the N ∈ H system of neighborhoods Σ in (??) is called projective limit of the family ( τ )τ T H ∈ and is denoted by ( =pr lim τ . N τ T H ∈

9 2. The space = pr limτ T τ is called nuclear iff τ T τ $ T such that N ∈ H ∀ ∈ ∃ ∈ the embedding operator i : % τ ",τ Hτ " → Hτ is of Hilbert-Schmidt type (or quasi nuclear).

Remark B.3 1. If the set T is countable, then =pr limτ T τ is called count- N ∈ H able Hilbert space.

2. The projective limit of Banach space is constructed in the same way where Hτ is replaced by a Banach space Bτ .

3. If T = N, then the system of base neighborhoods Σ in (??) is countable, therefore, turns into a Frec´ het space, i.e., a metrizable locally convex space. In N this case the topology in is generated by the following metric N ∞ 1 ϕ ψ ρ(ϕ, ψ) = | − |n . 2m 1 + ϕ ψ n=0 n " | − |

4. The convergence in is equivalent to the convergence in each n, n N0, N H ∈ (ϕj)j∞=1 converges to ϕ iff n N0 ϕj ϕ n 0, j . ⊂ N ∈ N ∀ ∈ | − | → → ∞ 5. Without lost of generality we can suppose that τ T % is a continuous ∀ ∈ Hτ → H0 embedding and ϕ ϕ , ϕ , τ T, 0 T. | |0 ≤ | |τ ∈ Hτ ∈ ∈ 2 Example B.4 Let T = τ = (τk)∞ , τk 1 , 0 := - (C) be given, i.e., { k=1 ≥ } H

∞ 2 2 0 = f = (fk)k∞=1, fk C f 0 = fk < . H % ∈ | | | | | ∞& "k=1 For each τ T deﬁne := -2(τ) by ∈ Hτ

∞ 2 2 τ = ϕ = (ϕk)k∞=1, ϕk C ϕ τ = ϕk τk < , H % ∈ | | | | | ∞& "k=1 with the following scalar product

∞ (ϕ, ψ) := ϕ ψ τ , ϕ, ψ -2(τ). τ k k k ∈ "k=1 2 It is not hard to show that the family (- (τ))τ T satisfies (DE) and (D) above and, ∈ 2 hence, we obtain a nuclear space := τ T - (τ) = C0∞, where C0∞ is the space N ∈ of infinite sequences with only a finite number of entries different from zero, e.g., ϕ = ( (ϕ1, ϕ2, . . . , ϕn, 0, 0, . . .).

10 Remark B.5 The projective limit of Hilbert spaces appear usually in the following form: given a linear space E and a family of scalar products ( , )n n N0 com- { · · | ∈ } patibles, i.e., if (f )∞ E is a convergent sequence to zero in the norm and is n n=1 ⊂ | · |n Cauchy in , then also it converges to zero in . Denote by the completion | · |m | · |m Hn of E with respect to and = ∞ . Since E , then is dense in each | · |n N n=0 Hn ⊂ N N n, n N0. We can always assume that the family of norms are increasing, i.e., H ∈ ( . (B.2) | · |0 ≤ | · |1 ≤ · · · | · |n ≤ | · |n+1 ≤ · · · For each τ T we can construct the following rigging of Hilbert spaces ∈

τ 0 τ , H− ⊃ H ⊃ H where τ is the dual of τ with respect to 0, cf. Appendix ??. H− H H

Theorem B.6 Let = τ T τ be given. Then we have N ∈ H ( $ = τ . N H− τ T #∈ Proof. The proof of this theorem can be founded in (?). !

Remark B.7 Hence we have constructed the following chain of spaces

$ = τ τ 0 τ = τ N H− ⊃ H− ⊃ H ⊃ H ⊃ N H τ T τ T #∈ $∈ which we abbreviate by $ N ⊃ H0 ⊃ N and is called Gelfand triple.

Remark B.8 The paring between $ and is determined by the scalar product of N N 0, i.e., if Φ $ and ϕ , then ϕ, Φ = (ϕ, Φ)0 C. In fact Φ τ , for H ∈ N ∈ N ( ) ∈ ∈ H− some τ T and ϕ , T which implies that the paring between $ and is ∈ ∈ Hτ ∀ ∈ N N determined by the paring between τ and τ . H− H 2 2 1 Example B.9 In the same conditions as in Example ?? we obtain that - (τ)$ = - (τ − ) 1 1 (here if τ = (τk)k N, then τ − = (τk− )k N) and as a consequence we have $ = 2 1 ∈ ∈ N τ T - (τ − ) = C∞. ∈ !

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