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Why is this property true of sets/indicator functions? (A page from my textbook)

  1. Mar 27, 2012 #1
    I can't for the life of me figure out why the underlined sentence in red is true:

    lebesguelinear.png

    Could someone clarify?
     
  2. jcsd
  3. Mar 27, 2012 #2

    LCKurtz

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    That page doesn't tell us what ##E## or ##S_0(E)## represent. Nor what the canonical representation of ##\phi \in S_0(E)## is.
     
  4. Mar 27, 2012 #3
    My apologies,

    ##E## is a measurable set.

    ##S_0(E)## is the set of simple functions on ##E##.

    The canonical representation of ##\phi \in S_0(E)## is a linear combination of characteristic functions ##\phi = \sum_{i=1}^na_i\mathcal{X}_{E_i}##.
     
  5. Mar 27, 2012 #4

    LCKurtz

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    I'm guessing there is more to the definition of canonical representation. Maybe you are given that the ##A_i## are disjoint with each other and ##B_j## similarly. And if ##\cup_i A_i = E## and ##\cup_j B_j = E## that would explain those equations.
     
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