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

  • Thread starter jdinatale
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I can't for the life of me figure out why the underlined sentence in red is true:

lebesguelinear.png


Could someone clarify?
 

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  • #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.
 
  • #3
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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.
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}##.
 
  • #4
LCKurtz
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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}##.
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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