(adsbygoogle = window.adsbygoogle || []).push({}); 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 χ_{E}defined by

χ_{E}(x) = 1 if x ε E; 0 if x [STRIKE]ε[/STRIKE] E

is called thecharacteristic functionof E. A linear combination

ρ(x) = Ʃa_{i}χ_{Ei}(x)

is called asimple functionif 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 {a_{1}, ..., a_{n}} the set of nonzero values of ρ, then

ρ = Ʃa_{i}χ_{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.

3. 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.

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# Homework Help: RThe canonical representation phi (measure theory) (Royden)

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