数理解析研究所講究録 1452 巻 2005 年 195-205

185

Another Interpretation of Measurable Norms andCylindrical Measures.
Keiko Harai
Graduate School of Humanities and Sciences, Ochanomizu University

1

Introduction

Gross [5] presented a new notion “measurable norm” , which was the start of thesplendid and fantastic theory of “abstract Wiener spaces”. This gives a condition
satisfying that the Gauss cylindrical measure extends to a measure. In 1971,
Feldman-LeCam [3] defined another notion of measurable norms. If the norm is continuous, then this is a necessary and sufficient condition for satisfying that a generalcylindrical measure extends to a measure.
$\mathrm{D}\mathrm{u}\mathrm{d}\mathrm{l}\mathrm{e}\mathrm{y}arrow$

Badrikian and Chevet [1] offered the following question:“Do these notions of two measurable norms coincide with each other for every cylindricalmeasure?”In 1984 Maeda [15] solved this problem negatively. These two notions are very close,
but subtly different.
Many mathematicians noted conditions involved with these two measurable norms
and investigated them. They gave similar conditions to these measurable norms. In
Section 3, we give seven conditions approximating to measurable norms. And we show

the relation of seven conditions.

We introduce a new definition of a Gauss cylindrical measures which was introducedby Baxendale [2]. There we compare it with the Gauss cylindrical measure defined in
the original sense. The Gauss cylindrical measures introduced by Baxendale containones without variance, so that they extends the original ones. We study Baxendale’s
Gauss cylindrical measures in detail.

2

Preliminaries

Let $X$ be a locally convex Hausdorff topological vector space over the real field, $X’$ its topological dual, -, the natural pairing between $X$ and $X’$ and $B(X)$ thebe a finite system of elements of $X’$ . Then by
Borel a-algebra of $X$ . Let, $(x,\xi_{n}))$ .
mapping onto the vector $((x,\xi_{1})$ ,we denote the operator from $X$ into
, $\xi_{n}\in X’$ andA set $Z\subset X$ is said to be a cylindrical set if there are ,$\mathbb{R}$$($—$\cdot)$$\{\xi_{1}, \ldots ,\xi_{n}\}$$\mathbb{R}^{n}$$x$$\ldots$$\xi_{1}$$\ldots$$B\in B(\mathbb{R}^{n})$196such that Z $=—-1(B)$ . Let

A map 72 from$\mathrm{C}_{X}$$\mathrm{C}_{X}$denote the collection of all cylindrical sets of X.

into $[0, 1]$ is called a cylindrical measure if it satisfies the following

conditions:(1) $\mu(X)=1$ ;(2) Fkstrict to a a-algebra of cylindrical sets which are generated by a fixed
finite system of functional. Then each such restriction is countably additive.
$\mu$

By putting
Borel probability measures.

$\mu_{\xi_{1},\ldots,\xi_{n}}(B)=\mu(_{-}^{--1}-(B))$

each cylindrical measure$\mu$

defines a family of

Next we introduce two kinds of measurable norms defined on a Hilbert space.
. Let $F$ be the partially
Let $H$ be a real separable Hilbert space with norm
ordered set of all finite dimensional orthogonal projections of $H$ and $FD(H)$ the familyof all finite dimensional subspaces of H. $P>Q$ means $PH\supset$ $QH$ for $P$, $Q\in F$ . Alsoa subset $E$ of $H$ of the following form is a cylindrical set, $E=\{x\in H;Px\in F\}$ , where$P\in F$ and $F$ is a Borel subset of $PH$ .
$|\cdot|=\sqrt{\langle\cdot,\cdot\rangle}$

First, we define the canonical Gauss cylindrical measure and two measurable norms.

Definition 2.1 The canonical Gauss cylindrical measure is a cylindrical measure
into $[0, 1]$ defined as follows:from$\gamma$$\mathrm{C}_{H}$If $E=\{x\in H;Px\in F\}$ , then$\gamma(E)=(\frac{1}{\sqrt{2\pi}})^{n}\int_{F}e^{-\frac{|x|^{2}}{2}}dx$,

where $n=\dim PH$ and $dx$ is the Lebesgue measure on $PH$ .
Remark 2.2 If $H$ is an infinite dimensional space, thenis not$\sigma$-additive. In general, we denote by$\gamma$

is finitely additive, but$\gamma^{t}(Z)=(\frac{1}{\sqrt{2\pi t}})n\int_{F}e^{-\frac{|x|^{2}}{2\mathrm{t}}}dx$

the Gausscylindrical measure with parameter $(0<t<\infty)$ .
is the canonical Gauss cylindricalmeasure. In this paper, we denote the canonical Gauss cylindrical measure by .
$t$$\gamma^{1}$$\gamma$

We define two measurable norms.

Definition 2.3 A semi-norm ||.|| in H is said to be 7- $(G)m$ easurable iffor every
and P $\in F$ .
there exists $P_{0}\in F$ such thatfor$\gamma(\{x\in H_{\}}.

||Px||>\epsilon\})<\epsilon$$\epsilon>0$,$\forall P[perp] P_{0}$This concept was introduced by Gross [7] in 1962. It was the starting point of thesuccessive research concerning the abstract Wiener space. In Definition 2.3, we canreplace with which is any cylindrical measure defined on $H$ . In such a case we say
thatis. We can also redefine the above concept as follows:$\gamma$$||\cdot$

$||$

$\mu$

$\mu-(G)\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$We say thatsuch that$H;||x||\leq\epsilon\}$ and$||\cdot||$, there exists $G\in FD(H)$if for every
whenever $F\in FD(H)$ and $F[perp] G$ , whereis the orthogonal complement of $F$ .
is$\mu-(G)\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$$\mu(N_{\epsilon}\cap F+F^{[perp]})\geq 1-\epsilon$$F^{[perp]}$$\epsilon>0$$N_{\epsilon}=\{x\in$1S7

Definition 2.4 A semi-norm . is said to be $\mu-(D)$ measurable if for every $>0$
whenever $F\in FD(H)$ andthere exists $G\in FD(H)$ such that
$F$ .
$H$
$F[perp] G$
ontois the orthogonal projection of, there$||$$||$$\epsilon$$\mu(P_{F}(N_{\epsilon})+F^{[perp]})$ $\geq 1-\epsilon$$P_{F}$

This was introduced by Dudley-Feldman-LeCam [3] in 1971.
They showed that these two measurable norms are equivalent for the canonical Gausscylindrical measure [3].

be a continuous serni-norm in a Hilbert space $H$ , andTheorem 2.5 ([3]) Let
the canonical Gauss cylindrical measure on H. Then the following statements areequivalent.
$||\cdot||$$\gamma$

(i)
(ii)

$)$

$||\cdot \mathrm{f}|$$||\cdot||$

is $\gamma-(D)measurable$ .

is$\gamma-(G)$measurable.

Let $E$ be the completion of $H$ with respect to the norm
is -measurable, then the triple
map of $H$ into $E$ . If
Wiener space.
$||\cdot$

$||$

.

$||$

$(\mathrm{z}, H, E)$$||$$\gamma$and the inclusion
is called an abstract$\mathrm{i}$be a continuous norm defined on a Hilbert space $H_{l}\mu a$
Theorem 2.6 ([2]) Let. Then thecylindrical measure on $H$ , and $B$ the completion of $H$ with respect tofollowing are equivalent.
$||\cdot$$||$$||\cdot$(i)$||\cdot||$(ii)$\mathrm{i}(\mu)$$||$

is $\mu-(D)$ measurable.

, whereis the image ofextensible to a measure.
$i(\mu)$under the map , is countably additive and is
$i$

$\mu$

It is easy to see that (GQmeasurability implies
false generally ([15], this is the 1984-Example).

$(D)\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}i\mathrm{t}\mathrm{y}$. But the converse is

We define the characteristic functions and the continuity of cylindrical measures.

Definition 2.7 The characteristic function$\phi$

of a cylindrical measure$\mu$

on H is de-fined by$\phi(\xi)=\int_{H}e^{\tilde{\iota}\langle\xi,x\rangle}\mu(dx)$where $\xi\in H$ .
We define the continuity of cylindrical measures.

Definition 2.8 Let be a cylindrical measure on H. The
if the characteristic function of is continuous
$\mu$

$\mu$

$\mu$

is said to be continuous

188

We introduce the following proposition. It is useful to check the continuity ofcylindrical measures.
Proposition 2.9 ([12]) Let be a cylindrical measure on H. Then the following are
equivalent.
is continuous.
(i)tends to zero, then we have(ii) if the generalized sequence
$\mu$

$\mu$

$\{\xi_{\alpha}\}\subset H$$\lim_{\alpha}\mu_{\xi_{\alpha}}([-\sigma, \sigma])=1$

for some (each)$\sigma>0$.

. Then it follows thatweakly converges to$(i\dot{x}i)$

Suppose3

Measurable norms and seven conditions$\lim_{\alpha}\xi_{\alpha}=0$$\mu_{\xi_{\alpha}}$$\delta_{0}$.

In 1980’s, Maeda compared two measurable norms (Gross’ and D.F.L.’s sense
respectively). She considered only two measurabilities but not other conditions. However there exist several conditions approximating to these measurabilities. In this section, we treat those conditions and research the relation between them.
In the following theorem, Baxendale showed that (i), (ii), (iii) and (vi) are equivalent for the canonical Gauss cylindrical measure. We have thatfor general cylindrical measures by a similar method. Condition (iv) was given by Yan
for any reflexive Banach space $B$ . After that,and Gong. Yan proved that
Gong showed it for an arbitrary Banach space $B$ . The equivalence of (v) and (vi) was
given in Theorem 2.7. The following theorem is a gathering of the above.

$(i)\Rightarrow(ii)\Rightarrow(i\mathrm{i}i)\Rightarrow(\mathrm{i}v)$$(iv)\Rightarrow(v\mathrm{i}i)$, $\mu a$
Theorem 3.1 Let $H$ be a real separable Hilbert space with nor
cylindrical measure on $H$ ,a continuous norm defined on $H_{f}B$ the completion of$H$ with respect toand the inclusion map from $H$ into B. Moreover, let $Y$ be the$B’$bidualof with weak’-topology ( , Bf) and be the inclusion map from $H$ into,
Y. Then the following seven conditions satisfy the relations:$rm|\cdot|=\sqrt{\langle\cdot,\cdot\rangle}$$||\cdot||$$||\cdot||$

$i$

$B$

$\mathrm{a}$$j$$\mathrm{B}\mathrm{r}/$$(i)\Rightarrow(ii)\Rightarrow(iii)\Rightarrow(iv)$$(i)\Rightarrow(i\mathrm{i})\Rightarrow(v)\Leftrightarrow(vi)\Rightarrow(vii)$

If.

is continuous (this means that the characteristic function of is continuous on.
, then the following conditions satisfy the relations: (iii)
and
$\mu$

$\mu$

$H)$

$\Rightarrow(vi)$(i) For an $y\epsilon>0$ there exists $N$ A
that $n>m\geq N$ implies$\mathrm{N}$, where$\mathrm{N}$$(\mathrm{i}v)\Rightarrow(v\mathrm{i}i)$

is the set of all natural numbers, such$\mu(\{x\in H;||P_{n}x-P_{m}x||>\epsilon\})<\epsilon$

for every sequencesuch thatto the identity map (we write it).
(ii)
is a $\mu-(G)$ measurable norm.
(iii) There eists a sequence
with
$>0$
cvny
there exists
such that$\{P_{n}\}\subseteq F$$I$$P_{n}<P_{n+1}(\forall n\in \mathrm{N})$and$P_{n}$

converges strongly$P_{n}\nearrow I$$||\cdot||$$\{P_{n}\}\subset F$$\epsilon$$P_{n}\nearrow I$

which has the property that for$N\in \mathrm{N}$$\mu(\{x\in H;||P_{n}x-P_{m}x||>\epsilon\})<\epsilon$

1afor every $n>m\geq N$ .

There exists a sequence $\{P_{n}\}\subset F$ withsuch thatand
any $>0$ there exist$P_{n}\nearrow I$$(\iota^{r}v)$, which has the properry that for$n_{\epsilon}\in \mathrm{N}$$N_{\epsilon}\in \mathrm{N}$$\epsilon$$\mu(\{x\in H,\cdot\sup_{1\leq k\leq n}||P_{k}x||>N\})<\epsilon$.
and every
is a $\mu-(D)measumble$ no .
(v)
) $(i.e. \mu\circ i^{-1})$ is extensible to a measure.
(vi)(vii) $j(\mu)$ is extensible to a measure.

for every$n\geq n_{\Xi}$$N\geq N_{\epsilon}$$rm$$||\cdot||$$\mathrm{i}\{\mu$The definition of Gauss cylindrical measures
introduced by Baxendale

4

The comparison between Gauss cylindrical measures by Baxendale and the original one was researched in [26]. Here we concretely construct a cylindrical measure on
using the same a as we introduced in Section 4, and we denote it by . We mention
later. It is a Gauss cyhndrical measure in the sense of Baxendale
the definition of
but it is not the original Gauss cylindrical measure. In this section, we study sevenconditions in Theorem 3.1 comparing and .
$\ell^{2}$$\gamma_{\mathrm{a}}$$\gamma_{\mathrm{a}}$$\gamma$$\gamma_{\mathrm{a}}$First we define Gauss cylindrical measures by Baxendale.

Definition 4.1 ([2]) Let E be a separable Banach space.
(a) A Borel probability measure on is Gaussian if either$\mathbb{R}$$\lambda$(i) A$=\delta_{0}$or(ii) There exists$t$$>0$ such that$\lambda(B)=\frac{1}{\sqrt{2\pi t}}\int_{B}e^{-\frac{|x|^{2}}{2\theta}}dx$

for$B\in B(\mathbb{R})$.

(b) A cylindrical measure is a Gauss cylindrical measure if every one-dimensional) is Gaussian on
distributionof ( $i.e$ . the image of under :$E$ is a Borel probability measure on $E$ which restricts to(c) A Gauss measure on$\mu$$\mu_{\xi}$$Earrow \mathbb{R}$

$\xi$

$\mu$

$\mu$

$\mathbb{R}$$a$Gauss cylindncal measure.

Definition 4.2 Let $E$ be a separable Banach space and a Gauss cylindrical measure
the characteristic function of . Let $L(E, E’)$ be the family of all
andonbounded linear operators from $E$ into $E’$ . We say that has a variance if there is$;\in E’$ . Hereself adjoint $A\in L(E,E^{J})$ , such that $\phi(\mu,\xi)=\exp\{-(\xi,A\xi)/2\}$ for all (is called self adjoint if ( ,Aq) ( ,A4) for any , $\eta\in E’$ .
$\mu$

$E$

$\phi(\mu, \xi)$

$\mu$

$\mu$

$A$

$a$

$A$

$=$

$\xi$

Let a be an element of$(\mathrm{a}, e_{\alpha})=0$$\gamma_{\mathbb{R}}$such thatfor $n=1,2$ ,for$\ell^{2^{*}}$$(\mathrm{a}, e_{n})=1$

Let

$\xi$

$\eta$$\ldots$}$e_{\alpha}\in J\backslash \{e_{n}\}_{n=1,2},\ldots\cdot$

be the canonical Gauss measure on$\mathbb{R}$,$\gamma_{\mathrm{a}}^{*}$

the measure induced by a function$\mathfrak{g}$

200

$f$

:

$\mathbb{R}arrow(\ell^{2})^{*}(tarrow t\mathrm{a})$measure$\gamma_{\mathrm{a}}$on$\ell^{2}$. i.e. 7: is the image measure of$\gamma_{\mathbb{R}}$under . Define a cylindrical$f$

as follows;For$Z=\{x\in(\ell^{2})^{*}; ((x,\xi_{1}), \ldots , (x, \xi_{n}))\in D\}$and$\tilde{Z}=\{x\in\ell^{2};(\langle x, \xi_{1}\rangle, \ldots, \langle x,\xi_{n}\rangle)\in D\}$.
,and
, where ,Baxendale, and is not the
of
in
the
sense
Then
is a Gauss cylindrical measureand doescanonical Gauss cylindrical measure. Further, it does not coincide withnot have a variance.
Put$\gamma_{\mathrm{a}}(\tilde{Z})=\gamma_{\mathrm{a}}^{*}(Z)$$\xi_{1}$$\xi_{n}\in\ell^{2}$$D\in B(\mathbb{R}^{n})$$\ldots$$\gamma_{\mathrm{a}}$$\gamma^{t}$

We have the following results forby using a similar method in the case of$||\cdot||_{1}$and$\mu_{\mathrm{a}}$$||\cdot||_{4}$.

and$\gamma_{\mathrm{a}}$. These results are shownis measurable for canonical Gauss cylindrical measures.
It is well-know thatis measurable for canonical Gauss cylindrical meaAnd we do not know whether
with respect toaresures. We have that this two norms . and
in
the
sense
of
Baxendale.
cylindricalameasurewhich is Gauss$||\cdot$$\mathrm{n}$$||_{4}$$||\cdot||_{1}$$||$$||\cdot||_{4}$$||_{1}$$(D)\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$$\gamma_{\mathrm{a}}$Theorem 4.3 ||.

$||_{1}$

Theorem 4.4 ||.

$||_{4}$is$\gamma_{\mathrm{a}}-(D)$measurable.

is$\gamma_{\mathrm{a}}-(D)$measurable.

The notion of measurable norms by Gross and the notion of measurable norms by
D.F.L. are equivalent for canonical Gauss cylindrical measures. We show the existenceof a Gauss cylindrical measure in the the sense of Baxendale that the notions of these
two measurability do not coincide.
Theorem 4.5 ||.

$||_{2}$is$\gamma_{\mathrm{a}^{\wedge}}(D)$measurable.

Proof. Let $E$ be the completion ofwith respect to the norm$E$$j’$inclusion map of
into . Denote by
the dual operator of . Let.
natural pairing$\ell^{2}$$||\cdot||_{2}$$\ell^{2}$$E’\mathrm{x}$$j$$(\cdot$, and the,be the$j$$\cdot)_{E}$$Earrow \mathbb{R}$$E’arrow j’(\ell^{2})’\simeq \mathit{1}^{2}arrow Ej$, the
To prove that the norm
, it suffices to show thatismap,
image ofunder theis -additive on.
Claim, a vanishes on $j’(E’)$ .
Proof of claim. Suppose $y\in E’$ is given. We have to show that $(\mathrm{a}, j’(y))=0$ . Since,, we may assume
far all
1s of the form
$A_{N}\in$where , A2, . . . ,R.
Now define a sequenceinby$||\cdot$$\gamma_{\mathrm{a}}$$(\mathrm{a}, e_{\alpha})=0$$||_{2}$$j$$j(\mu_{\mathrm{a}})$$\gamma_{\mathrm{a}}-(D)\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{b}1\mathrm{e}$$(E,\mathrm{C}_{E})$$\mathrm{c}\mathrm{r}$$e_{\alpha}\in J\backslash \{e_{n}\}_{n=1,2},\ldots$$\mathrm{J}\sim’(y)$$A_{1}$$\{x^{m}\}_{m=1,2},\ldots$$\ell^{2}$$\sum_{n=1}^{N}A_{n}e_{n}$

201

.

$x^{1}=e_{1}$,$x^{2}=e_{1}+e_{2}+e_{\mathit{3}}$,$x^{m}=e_{1}+\ldots+e_{2m-1}$.
.

,$\cdot$for all $m\geq N$ .
for $m\geq(k+1)/2$ , so$(y,j(x^{m}))_{E}=\sum_{n=1}^{N}A_{n}$for allMoreover, since $\langle j’(y), x^{m}\rangle=(y,j(x^{m}))_{E}$ , we obtainThen$\langle j’(y), x^{m}\rangle=\sum_{n=1}^{N}A_{n}$$\langle e_{k}, x^{m}\rangle=1$

Therefore,$m\geq N$ .

$\lim_{marrow\infty}$

But by constructions of. The assumption
Therefore,$\{\beta_{m}\}$$1/\beta_{2m-1}$$(y,j(x^{m}))_{E}= \sum_{n=1}^{N}A_{n}$and$U_{21}$$\beta_{2m-1}arrow$(1}.

we know thatoo as $marrow$ oo implies$\beta_{2m-1}x^{m}\in U_{2}$$\lim_{marrow\infty}j(x^{m})=0$, so thatas $marrow\infty$ .
$||x^{m}||_{2}\leq$$||x^{m}||_{2}arrow 0$

(2)

in E.

Thus by (17) and (18) we deduce that $\sum_{n=1}^{N}A_{n}=0$ , and hence $(\mathrm{a},j’(y))=$$\sum_{n=1}^{N}A_{n}=0$ . This completes the proof of our claim.

Let be the canonical map ofis the Dirac measure
that$E$ , so it is-additive on $(E,C_{E})$ .
on$(l^{2})^{*}$

$i$

, so
into $(E’)^{*}$ . Then our claim implies
is extendable toon $(E’)^{*}$ . Therefore,$\mathrm{i}(\mathrm{a})=0$$j(\gamma_{\mathrm{a}})$$\gamma_{0}=\delta_{0}$$\mathrm{i}(\gamma_{\mathrm{a}}^{*})$$\square \delta_{0}$$\sigma$

Theorem 4.6 ||.

$||_{2}$

is not $\gamma_{\mathrm{a}}-(G)measurable$ .

Proof. It suffices to show that there exists a number
and$G\in FD(\ell^{2})$ there exists $F\in FD(\ell^{2})$ satisfying$F[perp] G$$\epsilon_{0}>0$

Let $G$ be an arbitrary finite dimensional subspace of , and
whereof . Then each is of the form:
matrixfollowing
have
the
.
..
Then
we
$i=1,2$ , .
$\ell^{2}$$G$$\xi^{j}=\sum_{i=1}^{\infty}\alpha_{\overline{l}}^{j}e_{i}$$\xi^{j}$such that for every$\gamma_{\mathrm{a}}(\epsilon 0U_{2}\cap F+F^{[perp]})<1-\epsilon_{0}$.

be a CONS, and
for $j=1,2$ ,$\{\xi^{j}\}j=1,2,\ldots,n$$\alpha_{i}^{\mathrm{J}}\in \mathbb{R}$$\ldots$$n$$A$$A=\{$$\alpha_{n+m}^{1}\alpha_{n+m}^{n}.\cdot.)$$\alpha_{n}^{1}\alpha_{n}^{n}..\cdot$$\alpha_{1}^{n}\alpha_{1}^{1}..\cdot$$.-\cdot.-\cdot..$,.

where $m$ is chosen such that rank $A=n$ . Suppose $N>n+m$ . Then the next equation.
has its solution in$\mathbb{R}^{n+m}$202$A\ovalbox{\tt\small REJECT}$.

$x_{n+m}x_{n}x_{1}..\cdot.\cdot.)=(\begin{array}{l}-\alpha_{2N+1}^{1}\vdots-\alpha_{2N+\mathrm{l}}^{n}\end{array})$(3)

as $iarrow$ oo for $j=1$ , 2, , . Therefore,
By the construction we know thatfor every $>0$ , we may choose a positive integer $N(>n+m)_{\backslash },$ $N$ sufficiently large,, . . . , $x_{n+m}=\eta_{n+m}$ satisfying
such that the equation (19) has a solution$\alpha_{f}^{j}arrow \mathrm{o}$$n$$\ldots$$\delta$$x_{1}=\eta_{1}$.

(4)$1\leq\iota\leq n+m\mathrm{m}\mathrm{a}\mathrm{x}|\eta_{l}|<\mathit{5}$Now choose a number$\delta$$>0$ in (20) such that$\frac{\eta_{1}+\eta_{2}+...\cdot.+\eta_{n+m}+1}{(\eta_{1})^{2}+(\eta_{2})^{2}+.+(\eta_{n+m})^{2}+1}>\frac{1}{2}$(5).

. . . $+\eta_{n+m}e_{n+m}+e_{2N+1}$ and $F$ be the one dimensional subspace ofLet, , so that $F[perp] G$ .
for $j=1$ , 2,generated by .
.
, whereis the Hilbert norm of , thenPut$\tau=\eta_{1}e_{1}+$$\ell^{2}$$\tau$$\langle\tau,\xi^{j}\rangle=0$$\ldots$$n$$l^{2}$$|$$|$$\phi=\frac{\tau}{|\tau|}$$( \mathrm{a}, \phi)=\frac{(\mathrm{a}\tau))}{|\tau|}=\frac{\eta_{1}+\eta_{2}.+\ldots+\eta_{n+m}+1}{(\eta_{1}^{2}+..+\eta_{n+m^{2}}+1)^{\frac{1}{2}}}$

.

.
. Put
then
Let
. Suppose this is not the case. Then
Now we show thatand $Y\in$may be expressed as $(\mathrm{a}, \phi)\phi=X+Y$ , where
implies that
$Y$
$X$
$X$
,
and
,
are of the form
. Since ,andand by (21) we have
, 2, . Then
for
where
$\epsilon_{1}=\int_{\{\mathrm{a},\phi]}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}dx$$0< \epsilon_{1}<\frac{1}{2}$$(\mathrm{a}, \phi)\phi\not\in$$\epsilon_{0}=\min(\epsilon_{1}, \frac{1}{6})$$(\mathrm{a}, \phi)\phi\in\epsilon_{0}U_{2}$$\epsilon_{0}U_{2}$$X\in\epsilon 0\Gamma 2$$(\mathrm{a}, \phi)\phi$$Y\in\ell^{2}$$\epsilon_{0}B_{1}$$X_{i},Y_{i}\in \mathbb{R}$$\mathrm{i}=1$$X= \sum_{i=1}^{\infty}X_{i}e_{i}$$\ldots$$Y= \sum_{i=1}^{\infty}Y_{\dot{2}}e_{i}$$( \mathrm{a}, \phi)\phi=\sum_{i=1}^{\infty}(X_{i}+Y_{i})e_{\dot{\mathrm{t}}}$$X_{2N}+Y_{2N}=0$

and$X_{2N+1}+Y_{2N+1}= \frac{\eta_{1}+\eta_{2}+.\cdots+\eta_{n+m}+1}{\eta_{1^{2}}+..+\eta_{n+m}^{2}+1}>\frac{1}{2}$The fact$X\in\epsilon_{0}\Gamma_{2}$.

implies that $X_{2N}=X_{2N+1}$ . Therefore,$| \frac{\eta_{1}+\eta_{2}+.\cdots+\eta_{n+m}+1}{\eta_{1}^{2}+..+\eta_{n+m}^{2}+1}-Y_{2N+1}|=|X_{2N+1}|=|\mathrm{Y}_{2N}|<\epsilon_{0}\leq\frac{1}{6}$On the other hand by (21),,$| \frac{\eta_{1}+\eta_{2}+.\cdots+\eta_{n+m}+1}{\eta_{1}^{2}+..+\eta_{n+m^{2}}+1}-\mathrm{Y}_{2N+1}|>\frac{1}{2}-|Y_{2N+1}|\geq\frac{1}{3}$203

and since $N$ is sufficiently large, we reach a contradiction. Therefore we have$\epsilon_{0}U_{2}$.

By$(\mathrm{a}, \phi)\phi\not\in\epsilon_{0}U_{2}$, we obtain$q\phi\not\in\epsilon 0U_{2}$$\gamma_{\mathrm{a}}(\epsilon_{0}U_{2}\cap F+F^{[perp]})$for any$=$$\leq$$q\geq(\mathrm{a}, \phi)$. Since$(\mathrm{a}, \phi)\phi\not\in$$(\gamma_{\mathrm{a}})_{\phi}=\gamma_{\mathrm{a}}\circ\phi^{-\mathrm{I}}$,$(\gamma_{\mathrm{a}})_{\phi}(\epsilon_{0}U_{2}\cap F)$

1–2$\int_{\{\mathrm{a},\phi)}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}dx$$=$$1-2\epsilon_{1}$$<$$1-\epsilon_{1}$$\leq$$1-\epsilon_{0}$,and the proof is complete.

$\square$

We conclude that Gauss cylindrical measures of Baxendale contain not only the
canonical Gauss cylindrical measure and but also another cylindrical measure. And
they do not have a particular property that conditions (i), (ii), (iii), (v) and (vi) in
Theorem 3.1 are equivalent. This property characterizes the Gauss cylindrical measuresin the original sense.
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