Chapter 7 Infinite product spaces

2 CHAPTER 7. INFINITE PRODUCT SPACES
(5) Hilbert cube [0, 1] N with product (Euclidean) topology.
(6) R n with Euclidean metric.
(7) R N with product Euclidean topology.
(8) L p (W, F , µ) with 1 apple p < • provided F is countably generated and µ is
s-finite. (This includes the ` p spaces with p < •.)
(9) ` • fails to be Polish but its subspace of converging sequences
with supremum metric is Polish.
c0 = x =(xn) 2 ` • (N) : xn ! 0
(7.1)
(10) C(X)—the set of all continuous functions on a compact metric space X is Polish
in the topology generated by the supremum norm.
(11) L(X, Y)—the space of bounded linear operators T : X ! Y where X and Y are
separable Banach spaces is Polish in the strong topology (the norm topology
generally fails this property). The metric is defined by
d(T1, T2) =Â 2
n 1
n T1( fn) T2( fn) Y
where ( fn) is a countable dense set in X.
(7.2)
(12) M1(W, F )—the set of all probability measures on a standard Borel space (W, F )
in the topology of so called weak convergence. The metric is defined by
d(µ, n) =Â 2
n 1
n Eµ( fn) En( fn) (7.3)
where ( fn) is a countable dense set in C(W)—which, as can be shown, is separable.
Note that, in such cases, this leads to a hierarchy of spaces: measures on
(W, F ), measures on measures on (W, F ), etc.
The reason why standard Borel spaces are good for us is that there are essentially
only three cases of them that we will ever have to consider:
Theorem 7.2 [Borel isomorphism theorem] Let (X, B(X)) be a standard Borel space.
Then there is a Borel isomorphism f : (X, B(X)) ! (Y, B(Y)) onto a Polish space Y
where
8
>< {1, . . . , n}, if |Y| = n,
Y = N,
>:
[0, 1],
if Y is countable,
otherwise,
(7.4)
and where B(Y) is the Borel s-algebra induced by the corresponding Polish topology.