t.LA#braiStrncturesofSubsets ( II. 4 in Driver ' start with a sample space r ( any set ) .
Definition : A collection Is 2- is a field if Gg 1. RE F m
2 .
**}y ( Ei )
If, instead of 3 , we have the stronger
3 '
. If { Ej 's E , is a countable set of events in F
, then Ej E F
.
is countable }
- it I } are o - fields over r
,
- f. Fi Bf. I
2 .
µ
Prep : Let Es t be any collection of subsets of s .
.
It is called the o - field g¥d by E .
Pf . 6 (E) = Mf 5- : I is a o - field over r et. EEF} .
H
de , ) -
) .
Thepxretotied Let x be a topological space leg . Euclidean space Rd ) .
The Borel o - field BCX) is the o - field generated by the open subsets in X .
B CX) = of open subsets of X}
Events in BK) are called Borel sets .
Fun Fact Pat : For X -- Rd , every open set
( u is a countable union of th
, D = G. A
d =L : BURJ = Glaub) : acb}. = of Ca, b) : a -b}
= a ca , bi : a -b}