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 .

TheMoraloftheStorypi@Y-sco.nFE28fIThism.ght 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

.