Measurement

To keep things simple , let's talk about measuringangle segments in the unit circle

$ = { z e Q : 12-1=1 }

O. .For a given subset E s we'd like

assign a number

PCE ) =

Propertiesoftleasurementsl. .Pls ) =

,p : 2$ →-

2.

If Ei,EzE 8. are-

then PIE,o =

2 ! If { Ejlj? , are-then IPL Ej)

-3.

If E, ,Ez e-Stare-

then PCE,) BLED

Theorem-

:

Proof . If E s $ and ut $ then I and

UE =

are congruent . i . by 133,ME ) = BLUE ) Fut $ .

Now,consider $0 IT :-. { G

" it: te Q } (countable)

$ IT = f equivalence classes in $

where z n w ⇒ 2- = UW for some UET } .

Choose exactly t representative element q from each

equivalence class , and let Io = { 9 } a $ be the collectionof all these representatives .

Claim : $ =

u

u #.

Contradicts ?The measurement function P

, satisfyingID

,L2 ' )

,133 is used daily in calculus !

PCE ) =

So how can it fail to exist ?

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 .

tftheSyp : 2$ → co

.

I ]

⑧

This might 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

.