RThe canonical representation phi (measure theory) (Royden)

[Jamin2112]

RThe "canonical representation phi" (measure theory) (Royden)

Homework Statement

I need some help understanding the canonical representation of phi as described on p. 77 of Rodyen's 3rd edition. I've transcribed it below for those of you who don't own the book.

Homework Equations

The dunction χ_E defined by

χ_E(x) = 1 if x ε E; 0 if x [STRIKE]ε[/STRIKE] E

is called the characteristic function of E. A linear combination

ρ(x) = Ʃa_iχ_Ei(x)

is called a simple function if the sets E_i are measurable. This representation for ρ is not unique. However, we note that a function ρ is simply if and only if it is measurable and assume only a finite number of values. If ρ is a simple function and {a₁, ..., a_n} the set of nonzero values of ρ, then

ρ = Ʃa_iχ_Ai(x),

where A_i = {x: ρ(x) = a_i}. This representation for ρ is called the canonical representation, and it is characterized by the fact that the A_i are disjoint and the a_i distinct and nonzero.

The Attempt at a Solution

So I understand the ρ(x): there are coefficients a_i corresponding to each of the measurable sets E_i, and for a given x the function ρ(x) simply sums up the a_i corresponding to the sets containing x. However, the definition of ρ, the "canonical representation," is eluding me.

Any help appreciated.

 

[tiny-tim]

Hi Jamin2112!

… However, we note that a function ρ is simply if and only if it is measurable and assume only a finite number of values. If ρ is a simple function and {a₁, ..., a_n} the set of nonzero values of ρ, then

ρ = Ʃa_iχ_Ai(x),

where A_i = {x: ρ(x) = a_i}. This representation for ρ is called the canonical representation, and it is characterized by the fact that the A_i are disjoint and the a_i distinct and nonzero.
…
So I understand the ρ(x): there are coefficients a_i corresponding to each of the measurable sets E_i, and for a given x the function ρ(x) simply sums up the a_i corresponding to the sets containing x. However, the definition of ρ, the "canonical representation," is eluding me.

essentially, ρ is a step function with n steps

the canonical representation is the obvious one which divides ρ into single-step functions, each with steps of different heights

 

[LCKurtz]

Take a simple example. Let $E_1=[0,2),\ E_2=(1,3]$. Let $\rho = 2X_{E_1} + 3X_{E_2}$. Draw a careful graph of that. You will see it takes on 3 different values, on the intervals $[0,1],\ (1,2),\ [2,3]$. The canonical representation uses these three disjoint sets, each on which the function is a different constant.

 

[Jamin2112]

Thanks, guys! That clarifies it.

Now could you explain to me what a Borel set is? My book doesn't say what a Borel set is ... it only gives a definition of the a collection of Borel sets:

"The collection β of Borel sets is the smallest σ-algebra which contains all of the open sets."

(Okay ... But what is a Borel set itself?)

 

[tiny-tim]

... But what is a Borel set itself?)

It's any set in "the smallest σ-algebra which contains all of the open sets" …

in other words, it's as defined at http://en.wikipedia.org/wiki/Borel_set …

In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement.​