Measurement
To keep things simple , let's talk about measuringangle segments in the unit circle
$ = { z e Q : 12-1=1 }
,'
For a given subset E s 8 we'd like to
assign a number
PCE ) = proportion of the circle c- Co, if
Propertiesoftleasurementsl. .Pls ) = I
,p : 2
'S→
2.
If Ei,Eze 8. aredt E
, oEz=¢then PIE
,o = ME
,) TIMED
2 ! If { Ejlj? , are dim EioEj=¢ tij .
then Pl Ej) = IE ME;)3.
If E, ,Ezsst are0¥
then PCE,) = BLED
theorem : ¥
Proof . If,@
Es $ and at $ then I and
¥; ," " e → WE = { uz : ZEE} -- -
are congruent . i . by 133,ME ) = BLUE ) Fut $ .
Now,consider $0 IT :-. { E
" it: te Q } (countable)
$ IT = f equivalence classes in $
*offchoice-
where z n w ⇒ z : UW for some UET } .
xactyrepresentative element q from each
equivalence class , and let # = { 91 a $ be the collectionof all these representatives .
Claim : is =
a
ate.
H $ '# u# t
/ Kj if u,#ET,
u,Ion u¥=¢
1. =p,
= Platefuls},:&,
Pluto) is, Erm# ={ If
.
Contradicts ?The measurement function P
, satisfyingID
,L2 ' )
,133 is used daily in calculus !
PCE ) = ¥±d° ↳ and phg inso how can it fail to exist ? any
old set.
The answer lies in an important subtlety :the definition of the Riemann integralonly works over " nice
" sets. The set Io
is not nice !
Much of this quarter will be spentextending the Riemann integral . BUTthere's only so far it can be extended .
[email protected] seem like a bad sign . . .
but it is actually a foundational truthfor Kolmogorov 's probability theory(that we now embark on developing l .
In short : we don't always havecomplete information about the world ,which means there may be some eventswe simply cannot assign probabilitiesto
.
As to the unmeasurable sets . . .
Banaoh-Tarski_Paadx ( 1942) or Rd for d> 3
Given anytwo subsets E
,F E 1123% ith
nonempty interior, there are finite partitions
E = E,u Ez U - - - U En
F = F,
U E U- -
- u Fn
such that Ej is congruent to Fj for lsjsn .
RobinsubngTheen ( 1947 )
If E is a solid ball in 1123,
and F is the disjoint ballsof the same radius
,then
Banach -Tarski works explicitlywith n = 5
.