The Bochner-Minlos theorem

Jordan Bell
May 13, 2014

1 Introduction

We take to be the set of positive integers. If A is a set and n, we typically deal with the product An as the set of functions {1,,n}A.

In this note I am following and greatly expanding the proof of the Bochner-Minlos theorem given by Barry Simon, Functional Integration and Quantum Physics, p. 11, Theorem 2.2.

2 The Kolmogorov extension theorem

If X is a topological space, and for mn the maps πm,n:XmXn are defined by


then the spaces Xn and maps πm,n constitute a projective system, and its limit in the category of topological spaces is X with the maps πn:XXn, where X has the initial topology for the family {πn:n} (namely, the product topology). We say that a function f:X depends on only finitely many coordinates if there is some n and some function g:Xn such that f=gπn. We denote by Cfin(X) the set of all continuous functions X that depend on only finitely many coordinates.

If (X,τX) is a noncompact locally compact Hausdorff space, write X˙=X{}, and let τ be the collection of all subsets U of X˙ such that either (i) UτX or (ii) U and XU is compact in (X,τX). One proves11 1 Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second ed., p. 132, Proposition 4.36. that (X˙,τ) is a compact Hausdorff space and that the inclusion map ι:XX˙ is a homeomorphism Xι(X), where ι(X) has the subspace topology inherited from X˙. Also, if fC(X) then there is some FC(X˙) whose restriction to X equals f if and only if there is some gC0(X) and some constant c such that f=g+c, in which case


We call X˙ the one-point compactification of X. For example, one checks that the one-point compactification of n is homeomorphic to Sn.

Theorem 1 (Kolmogorov).

Suppose that for each n, μn is a Borel probability measure on n, and that for any n and any Borel set A in n we have


equivalently, πm,n*μm=μn for mn. There is then a Borel probability measure μ on such that for any n and any Borel set A in n,


equivalently, πn*μ=μn.


Let X=˙, the one-point compactification of , and let X have the product topology, with which it is a compact Hausdorff space. For each n, if A is a Borel set in Xn, we define νn(A)=μn(An). This is a Borel measure on Xn. If gC(Xn), mn, and h=gπm,n, then


We define L:Cfin(X) in the following way. For fCfin(X), there is some n and some gC(Xn) such that f=gπn. We define


If hC(Xm) and f=hπm with mn, then h=gπm,n, giving


so the definition of L(f) makes sense.

It is straightforward to check that Cfin(X) is an algebra over . The algebra Cfin(X) separates points in X, and the constant map x1 belongs to Cfin(X); the latter fact tells us that there is no xX such that f(x)=0 for all fCfin(X). Therefore, the Stone-Weierstrass theorem22 2 Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second ed., p. 141, Corollary 4.50. tells us that Cfin(X) is dense in the Banach algebra C(X).

If fCfin(X) and f=gπn, then f=g, which is finite because Xn is compact. Because each νn is a probability measure,


showing that L:Cfin(X) is a bounded linear map, with L=1. Because Cfin(X) is dense in C(X), there is a bounded linear map Λ:C(X) whose restriction to Cfin(X) is equal to L, and that satisfies Λ=L=1. Moreover, if fCfin(X) satisfies f0, then it is apparent that L(f)0; we say that L is a positive linear functional. The fact that L is a positive linear functional implies that Λ is too. Because Λ:C(X) is a positive linear functional with Λ=1, by the Riesz-Markov theorem33 3 Walter Rudin, Real and Complex Analysis, third ed., p. 40, Theorem 2.14. there is a Borel probability measure ν on X such that


If A is a Borel set in with the product topology, define μ(A)=ν(A). μ is a Borel probability measure on .

Now that we have in our hands a Borel probability measure μ on it remains to verify that it does what we want it to do. ∎

3 Sequence spaces

For x,y and m, we define


and xm=x,xm1/2. We define 𝔖m to be the set of all those x for which xm<, and we take as granted that for each m, 𝔖m is a Hilbert space. For mn, let ιm,n:𝔖m𝔖n be the inclusion map. As

ιm,nxn2 = j=1j2n(ιm,nx)(j)2
= j=1j2mj2(n-m)x(j)2

the map ιm,n is a bounded operator. In fact, if m>n we now demonstrate that ιm,n is a Hilbert-Schmidt operator, and so compact, which is the conclusion of Rellich’s theorem. For i, define ei by


These ei are an orthonormal basis for 𝔖m, and

i=1ιm,nein2 = i=1j=1j2n(ιm,nei)(j)2
= i=1j=1j2n(j-mδi,j)2
= i=1i2ni-2m
= i=1i2(n-m).

Because m>n, this last expression is <. This shows that ιm,n:𝔖m𝔖n is a Hilbert-Schmidt operator.

For x𝔖m and λ𝔖-m, define

x,λ=j=1x(j)λ(j), (1)

and using the Cauchy-Schwarz inequality,

|x,λ| j=1|x(j)||λ(j)|
= j=1jm|x(j)|j-m|λ(j)|
= xmλ-m.

𝔖-m is thus the dual space of the Banach space 𝔖m. That is, as a vector space 𝔖m*=𝔖-m, but we shall be interested in 𝔖m* with the weak-* topology with which it is a locally convex space, rather than with the norm topology with which it is a Banach space.

Since ιm,n:𝔖m𝔖n is a bounded linear map for mn, the spaces 𝔖m and the maps ιm,n are a projective system of Banach spaces, and this projective system has a limit 𝔖 in the category of locally convex spaces. This limit 𝔖 is a Fréchet space. The duals 𝔖m* with the weak-* topology are locally convex spaces and constitute a direct system with the maps (ιm,n)*:𝔖n*𝔖m*, where (ιm,n)*(λ)=λιm,n for λ𝔖n*. This direct system has a colimit in the category of locally convex spaces which is equal to 𝔖* with the weak-* topology.44 4 Paul Garrett,, p. 15. As sets,


We also denote by , the dual pairing of 𝔖 and 𝔖*: for x𝔖 and λ𝔖*,


For any λ𝔖*, there is some m for which λ𝔖m*=𝔖-m. But if x𝔖 then x𝔖m, and so this dual pairing coincides with (1).

4 Positive-definite functions

If X is a vector space and Φ:X is a function, we say that Φ is positive-definite if for any positive integer r and for any x1,,xrX and c1,,cr, we have


If Φ is positive-definite, one proves that Φ(0)0, Φ(-x)=Φ(x)¯, and |Φ(x)|Φ(0).

If μ is a probability measure on (𝔖*,Cyl(𝔖*)), we define the Fourier transform of μ to be the function μ^:𝔖 defined by


because Lx:𝔖* and μ is a probability measure, this integral is finite. Using the dominated convergence theorem, one checks that μ^:𝔖 is continuous. It is apparent that μ^(0)=1. If x1,,xr𝔖 and c1,,cr, then

j,k=1rcjck¯μ^(xj-xk) = j,k=1rcjck¯𝔖*exp(-iλ(xj-xk))𝑑μ(λ)
= 𝔖*j,k=1rcjexp(-iλxj)ckexp(-iλxk)¯dμ(λ)
= 𝔖*(j=1rcjexp(-iλxj))(k=1rckexp(-iλxk))¯𝑑μ(λ)
= 𝔖*|j=1rcjexp(-iλxj)|2𝑑μ(λ)

so μ^:𝔖 is positive-definite.

5 Cylinder sigma-algebras

𝔖* is a topological space and so has a Borel σ-algebra. We shall now define a σ-algebra on 𝔖*, called the cylinder σ-algebra of S* and denoted Cyl(𝔖*), that is strictly contained in the Borel σ-algebra of 𝔖*. For x𝔖, define Lx:𝔖* by Lx(λ)=λ(x)=x,λ. We define Cyl(𝔖*) to be the coarsest σ-algebra such that for each x𝔖, the map Lx:𝔖* is measurable, where has the Borel σ-algebra. Since each Lx is continuous, Lx is measurable with respect to the Borel σ-algebra on 𝔖*, so Cyl(𝔖*) is contained in the Borel σ-algebra of 𝔖*; it is not obvious that the cylinder σ-algebra is strictly contained in the Borel σ-algebra. Unless we say otherwise, when we speak of measurable functions on 𝔖* or measures on 𝔖* we mean with respect to the cylinder σ-algebra.

6 Minlos’s theorem

In the following theorem we obtain a Borel probability measure μ on with the product topology. We denote by 𝔅 the Borel σ-algebra of . The collection 𝔅0={B𝔖:B𝔅} is a σ-algebra on 𝔖. We assert that 𝔅0Cyl(𝔖), and that Cyl(𝔖) does not contain the Borel σ-algbera of 𝔖, and thus that the restriction of μ to 𝔖 is not a Borel measure.

Theorem 2 (Minlos).

If Φ:𝔖 is positive-definite, continuous, and Φ(0)=1, then there is some probability measure μ on (𝔖*,Cyl(𝔖*)) such that Φ=μ^.


For MN, define πM,N:MN by


The Banach spaces N and the maps πM,N constitute a projective system in the category of locally convex spaces, with the limit , which is thus a Fréchet space, with the maps πN:N,


The dual maps πM,N*:(N)*(M)* are defined for λ(N)* by


(N)*=N and the maps πM,N* constitute a direct system, and their colimit in the category of locally convex spaces is denoted


=()*, and the maps πN*:(N)* satisfy


The function ΦN=ΦπN*:(N)* satisfies, for λ1,,λr(N)* and c1,,cr,


because λ1πN,,λrπN𝔖 and Φ:𝔖 is positive-definite. Furthermore, (ΦπN*)(0)=Φ(0)=1, and ΦN=ΦπN* is a composition of continuous functions so is itself continuous. Therefore, for each N we have by Bochner’s theorem that there is one and only Borel probability measure μN on N that satisfies ΦN=μ^N. If MN, for ξ(N)*=N we have

(πM,N*μM)(ξ) = Ne-iξxd(πM,N*μM)(x)
= Me-iξπM,N(x)𝑑μM(x)
= Me-iπM,N*(ξ)x𝑑μM(x)
= μ^M(πM,N*(ξ))
= ΦM(πM,N*(ξ))
= (ΦMπM,N*)(ξ)
= ΦN(ξ)
= μ^N(ξ).

Since (πM,N*μM)=(μN), it follows that πM,N*μM=μN. Therefore, the Borel probability measures μN satisfy the conditions of the Kolmogorov extension theorem, and so there is some Borel probability measure μ on such that πN*μ=μN. Now that we have our hands on the measure μ, one must prove that μ^=Φ.

Supposing that we have proved μ^=Φ, we now prove that μ(𝔖*)=1. Let ϵ>0. Φ:𝔖 is continuous at 0 and Φ(0)=1, and as 𝔖 has the locally convex topology induced by the family of seminorms m, there is some m and some δ>0 such that ymδ implies that |Φ(y)-1|ϵ. Suppose that y𝔖. On the one hand, if ym2δ2, then


On the other hand, if ym2>δ2, using |Φ(y)|Φ(0)=1 and so |ReΦ(y)||Φ(y)|1, we get


Therefore, for any y𝔖,


Then, for yN we have

ReΦ(πN*(y))1-ϵ-2δ-2πN*(y)m2. (2)

Let α>0, let qk=k-2m-2, and let σα,N be the measure on N with


Using exp(-x2)𝑑x=π, it is straightforward to check that σα,N is a probability measure on N. Furthermore, we calculate, using respectively xexp(-x2)𝑑x=0, exp(-x2)𝑑x=π, and x2exp(-x2)𝑑x=π2,

Nyiyj𝑑σα,N(y) = δi,jyj2k=1N(2παqk)-1/2exp(-yk22αqk)dyk
= δi,j(kj(2παqk)-1/2exp(-yk22αqk)𝑑yk)
= δi,jyj2(2παqj)-1/2exp(-yj22αqj)𝑑yj
= δi,jαqj.

and for xN, taking as known the Fourier transform of a Gaussian function,

Ne-ixy𝑑σα,N(y) = k=1Ne-ixkyk(2παqk)-1/2exp(-yk22αqk)𝑑yk
= k=1Nexp(-αqkxk22)
= exp(-α2k=1Nqkxk2).

Then, using Φ=μ^, the integral of the left-hand side of (2) over N with respect to σα,N is

ReNΦ(πN*(y))𝑑σα,N(y) = ReNexp(-iπN*(y),x)𝑑μ(x)𝑑σα,N(y)
= ReNexp(-iyπN(x))𝑑σα,N(y)𝑑μ(x)
= Reexp(-α2k=1Nqkxk2)𝑑μ(x)
= exp(-α2k=1Nqkxk2)𝑑μ(x).

The integral of the right-hand side of (2) over N with respect to σα,N is

N(1-ϵ-2δ-2πN*(y)m2)𝑑σα,N(y) = 1-ϵ-2δ-2Nk=1Nk2myk2dσα,N(y)
= 1-ϵ-2δ-2k=1Nk2mαqk2
= 1-ϵ-2δ-2αk=1Nk-2.

Combining these, (2) is


Taking N,

exp(-α2k=1qkxk2)𝑑μ(x)1-ϵ-2δ-2αζ(2). (3)


limα0+exp(-α2k=1Nqkxk2) = limα0+exp(-α2k=1Nk2(-m-1)xk2)
= {1x𝔖-m-10x𝔖-m-1,

so taking α0+, (3) yields


i.e. μ(𝔖-m-1)1-ϵ, and μ(𝔖*)μ(𝔖-m-1). That is, we have proved that for any ϵ>0 there is some m such that


which shows that μ(𝔖*)=1, i.e. that μ is a probability measure. ∎

7 References

There don’t seem to be many detailed presentations of the Bochner-Minlos theorem, or Minlos’s theorem, in the literature. Cf. Sazonov’s theorem. Doing some searching through books, the following look like they have enough details so that they are not worse than nothing, which is more than can always be said:

  • Helge Holden, Bernt Øksendal, Jan Ubøe and Tusheng Zhang, Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, second ed., Appendix A, pp. 257ff.

  • Takeyuki Hida, Brownian Motion, pp. 116ff., §3.2

  • I. M. Gelfand and N. Ya. Vilenkin, Generalized Functions. Volume 4: Applications of Harmonic Analysis

  • V. I. Bogachev, Measure Theory, vol. II, p. 124, Theorem 7.13.7

  • A. V. Skorokhod, Basic Principles and Applications of Probability Theory, p. 51, §2.4.4

  • N. Bourbaki, Integration II: Chapters 7–9, p. IX.100, §6, Theorem 4