From: baez@galaxy.ucr.edu (John Baez)
Subject: Re: path integral - sum over all paths?
Date: 26 May 2000 16:22:45 GMT
Newsgroups: sci.physics.research
Summary: Cylinder measures

In article <8glulu$4no$1@nnrp1.deja.com>,  <squark@my-deja.com> wrote:

>But I want to be bored and terrified with details! Sounds logical, but
>what is the exact function space? I guess it's the functions generated
>by the above "cylindric" ones (closed in a suitable way?)?

Exact functions space coming up!  Expect to be terrifically bored.

Let V be a real topological vector space - infinite-dimensional for 
maximum fun.

A "cylinder function" f: V -> C is a function of the form 

f(x) = F(l_1(x),...l_n(x))

where l_i: V -> R are continuous linear functionals and F: R^n -> C
is a Borel-measurable function.  
 
Let the *-algebra of bounded cylinder functions on V be denoted Cyl(V).
Cyl(V) is a normed *-algebra with the sup norm.  We can complete it to
a commutative C*-algebra Fun(V).  

A "cylinder measure" is a bounded linear functional m: Fun(V) -> C 
which is nonnegative in the sense that f >= 0 implies m(f) >= 0.  
In other words, it's a gadget that knows how to integrate cylinder
functions and suitable limits thereof.

One can easily check that any finite Borel measure on V gives a
cylinder measure on V, since Fun(V) is a sub-algebra of the bounded
Borel-measurable functions on V.  But not conversely - when V is 
infinite-dimensional!   

There are lots of slightly different ways to make these definitions,
by the way, and also lots of variations in terminology.  I'm sticking
fairly close to the definitions in my book with Segal and Zhou, though
that book uses the term "tame functions" instead of "cylinder functions",
and it uses a funky concept called an "integration algebra" here instead of
the (I hope!) more familiar concept of "C*-algebra".  Segal invented
both these concepts; the former concept is a nice abstraction of the 
bounded measurable functions on a measure space, just as the latter is
a nice abstraction of the bounded continuous functions on a topological
space.  

This reminds me of my youth, when I was in love with functional analysis....