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Why is the sum of two simple functions also a simple function?

  1. Apr 11, 2010 #1
    Last edited by a moderator: Apr 25, 2017
  2. jcsd
  3. Apr 11, 2010 #2

    LCKurtz

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    What is there to prove? The definition says a simple function is a linear combination of indicator functions of measurable sets. A sum of two of them is just a longer linear combination of indicator functions of measurable sets.
     
    Last edited by a moderator: Apr 25, 2017
  4. Apr 11, 2010 #3
    Sorry. I left something out.

    Suppose now the definition of a http://en.wikipedia.org/wiki/Simple_function#Definition" also requires the events [tex]A_k[/tex] to be mutually exclusive.

    How can we now show that the sum of two simple functions will also be simple?
     
    Last edited by a moderator: Apr 25, 2017
  5. Apr 11, 2010 #4

    LCKurtz

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    It looks equivalent to me to the following statement:

    Suppose:

    [tex] A = \cup_{i=1}^n A_i[/tex]

    where the Ai are measurable sets. Then A can be rewritten a

    [tex] A = \cup_{i=1}^n B_i[/tex]

    where the Bi are disjoint measurable sets. And this looks trivial to prove by induction.
     
    Last edited by a moderator: Apr 25, 2017
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