e consider the sequence of r.v.
and a triple
.
The
is the event space of
and
is the minimal
algebra
that makes all
measurable. We think of
as a consecutive sequence of trials as
increases. Hence,
is a "process" and
is a time parameter.
We use the notation
to represent the
algebra
containing the information available up to time
.
Hence,
is the minimal
algebra
that makes the family
measurable.
We use the notation
to represent the
algebra
containing the information that comes after the time
.
Hence,
is the minimal
algebra
that makes the the family
measurable.
We use the notation
to represent the minimal
algebra
that contains all of the
,
.
By the minimality of
we have
We use the notation
for the intersection


(Remote field)

Sometimes it is convenient to think of
as
the product
space


(Random walk space)

where each
is the probability space for
.
The
is the product measure consistent with the d.f.
on each
.
The
is a collection of infinite sequences of real
numbers:
The
denotes the
"shift":


(Shift)

The
is the
applied
times.
The
denotes the set of permutations of
integers from the range
.
A permutation
(and similarly
)
produces a mapping on
and on
according to the
rules


(Permutation)

Definition
(Invariant set) The set
is called "invariant" if
.
Definition
(Remote event). Any set
(see the formula (
Remote field
)) is called
"remote event".
Clearly, an invariant set is remote and a remote set is permutable.
