Solution to Homework #1, 36754
27 January 2006
Exercise 1.1 (The product
σ
field answers count
able questions)
Let
D
=
S
X
S
, where the union ranges over all countable sub
sets
S
of the index set
T
. For any event
D
∈ D
, whether or not a
sample path
x
∈
D
depends on the value of
x
t
at only a countable
number of indices
t
.
(a) Show that
D
is a
σ
field.
(b) Show that if
A
∈ X
T
, then
A
∈ X
S
for some countable subset
S
of
T
.
Cf. the proof of Theorem 29 in the notes.
(a): We must show that (i) Ξ
T
∈ D
, (ii)
A
∈ D ⇒
Ξ
T
\
A
∈ D
and (iii)
A
n
∈ D ⇒
n
A
n
∈ D
for any countable collection of sets
A
n
.
(i): Pick
S
=
{
t
}
for any
t
∈
T
, and take the base set to be Ξ, i.e, the base
set is
x
∈
Ξ
T
:
x
t
∈
Ξ
t
. Clearly, this set is Ξ
T
.
(ii): Fix
S
. Then for any
A
∈ X
S
,
Ξ
T
\
A
=
Ξ
T
\
A
×
t
∈
T
\
S
Ξ
t
=
(
Ξ
S
\
A
)
×
t
∈
T
\
S
Ξ
t
which is in
X
S
.
(iii): Take any countable collection of sets
A
n
∈ D
. For each such set, there
is a corresponding finite set of indices,
S
n
, for which
A
n
∈ X
S
n
. Let
S
=
n
S
n
.
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 Spring '06
 Schalizi
 Natural number, Kolmogorov Extension Theorem, Ionescu Tulcea Theorem

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