Theorem (von Neumann's big theorem on algebra maps):
Let
be Polish spaces that are part of the outer regular measure spaces
and
. Suppose that
is an algebra map such that for all
, there exists an
such that
. Then there exists
and
such that
and a bijective function
such that
- for
,
and
- for
,
, where
is any set such that
.
Proof: First, we write
,
where
is a nullset and each
is closed and of diameter
. Then write
and
. Then we write
,
where
is closed and each
has diameter
. Then set
to be a set such that
and
its closure.
Continuing this ping-pong game, we obtain nested sequences
and
dependent on different numbers of indices. Moreover, if we set
and
,
then
, so that for any given number of indices, the corresponding sequence of nested sets
resp.
covers almost all of
resp.
, dependent on whether the number of indices is even or odd. Define
and
. Then define a function
as follows: If
, then for any given number of indices
there exists one and only one
that contains
. Moreover,