Statistics 851 (Fall 2013)
September 13, 2013
Prof. Michael Kozdron
Lecture #5: The Borel Sets of
R
We will now begin investigating the second of the two claims made at the end of Lecture #3,
namely that there exists a
σ
algebra
B
1
of subsets of [0
,
1] on which it is possible to de±ne
a uniform probability.
Our goal for today will be to de±ne the Borel sets of
R
.T
h
ea
c
t
u
a
lc
o
n
s
t
r
u
c
t
i
o
no
ft
h
e
uniform probability will be deferred for several lectures.
Recall that a set
E
⊆
R
is said to be
open
if for every
x
∈
E
there exists some
°>
0
(depending on
x
)suchtha
tthein
te
rva
l(
x
−
°, x
+
°
)i
scon
ta
inedin
E
. Also recall that
intervals of the form (
a,b
)for
−∞
<a<b<
∞
are open sets. A set
F
⊆
R
is said to be
closed
if
F
c
is open. Note that both
R
and
∅
are simultaneously both open and closed sets.
If we consider the collection
O
of all open sets of
R
,thenitfo
l
lowsimmed
iate
lythat
O
is not
a
σ
algebra of subsets of
R
.(Tha
ti
s
,i
f
A
∈O
so that
A
is an open set, then, by de±nition,
A
c
is closed and so
A
c
/
.) However, we know that
σ
(
O
), the
σ
algebra generated by
O
,
exists and satis±es
O
°
σ
(
O
)
⊆
2
R
.Th
i
sl
ead
stothefo
l
low
ingde±n
i
t
ion
.
DeFnition.
The
Borel
σ
algebra
of
R
,w
r
i
t
ten
B
sthe
σ
algebra generated by the open
sets. That is, if
O
denotes the collection of all open subsets of
R
,then
B
=
σ
(
O
).
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 Fall '14
 Statistics, Topology, Probability, Empty set, Metric space, Open set, Closed set, ij

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