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RThe canonical representation phi (measure theory) (Royden)

  1. Apr 16, 2012 #1
    RThe "canonical representation phi" (measure theory) (Royden)

    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations

    The dunction χE defined by

    χE(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) = ƩaiχEi(x)

    is called a simple function if the sets Ei 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, ..., an} the set of nonzero values of ρ, then

    ρ = ƩaiχAi(x),

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

    3. The attempt at a solution

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

    Any help appreciated.
     
    Last edited by a moderator: Apr 16, 2012
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  3. Apr 16, 2012 #2

    tiny-tim

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    Hi Jamin2112! :smile:
    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 :wink:
     
  4. Apr 16, 2012 #3

    LCKurtz

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

    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.
     
  5. Apr 17, 2012 #4
    Re: RThe "canonical representation phi" (measure theory) (Royden)

    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?)
     
  6. Apr 17, 2012 #5

    tiny-tim

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

    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. ​
     
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