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F non-measurable but |f| and f^2 are

  1. Feb 12, 2017 #1
    1. The problem statement, all variables and given/known data

    Find a function in a measurable space that is non-measurable, but |f| and f2 are measurable.

    2. Relevant equations

    None.

    3. The attempt at a solution

    I am trying to understand the following answer to the problem:

    MMx7fFs.png
    (source: http://math.stackexchange.com/a/1233792/413398)

    I do not understand why for all subsets [itex]S \subseteq \mathbb{R}[/itex], we have $$(|f|)^{-1}(S)=(f^2)^{-1}(S)=X \text{ or } \varnothing$$
     
  2. jcsd
  3. Feb 12, 2017 #2
    Hi,

    ##f^2## or ##abs(f)## both send ##a## and ##b## to ##1##. So ##1## has ##a## and ##b## as antecedents, and any other real has no antecedent. So if ##S## contains ##1## it has ##X=\{a,b\}## as reciprocal image, and the empty set if it doesn't contain it.
     
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