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Homework Help: Simple question about measurable characteristic function

  1. May 24, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove that the characteristic function [itex]\chi_A: X\rightarrow R, \chi_A(x)=1,x\in A; \chi_A(x)=0, x\notin A[/itex], where A is a measurable set of the measurable space [itex] (X,\psi) [/itex], is measurable.


    2. Relevant equations
    a function [itex]f: X->R[/itex] is measurable if for any usual measurable set M of R, [itex]f^{-1}(M)[/itex] is measurable in [itex](X,\psi)[/itex]

    3. The attempt at a solution
    Obviously [itex]f^{-1}([0,1])=X[/itex], where the universal set X need not be a measurable set in a general measurable space [itex](X,\psi)[/itex], which only requires that the (uncountable) union of all measurable sets is X. But the book explicitly asked to prove for a general measurable space. What am I missing here? Thank you.
     
  2. jcsd
  3. May 24, 2012 #2
    [itex]X=A\cup A^{c}[/itex]
     
  4. May 24, 2012 #3
    But [itex]A^c[/itex] need not be measurable in a general measurable space, which is not necessarily a Borel field, only a [itex]\sigma[/itex]-ring whose union is X. Am I completely wrong?
     
  5. May 24, 2012 #4
    I've heard of sigma fields and sigma algebras before, but never sigma rings... Is that a variation in which measurability is not closed under complements? I would double check that in your book.
     
  6. May 24, 2012 #5
    Every book I've seen defines a measurable space as a set equipped with a Σ-algebra. What book are you using?
     
  7. May 25, 2012 #6
    This is the book I use
    http://books.google.com/books/about/Some_modern_mathematics_for_physicists_a.html?id=9PXuAAAAMAAJ

    The definition of general measurable space in this book
    Definition 7.1(3). Let X be a (universal) set and let psi be a sigma-ring on X which has the property that X is a (not necessarily countable) union of sets taken from the collection psi. Then the ordered pair (X, psi) is called a measurable space. The members of psi are referred to as the measurable sets of X.

    A example in the book is let X be an uncountable set and psi be all countable subset of X.

    This is a wiki page on sigma-ring:
    http://en.wikipedia.org/wiki/Sigma-ring
    In the last paragraph:
    "σ-rings can be used instead of σ-fields in the development of measure and integration theory, if one does not wish to require that the universal set be measurable."

    In the same book there is this problem:
    Let (X, psi) be a general measurable space. Show that the characteristic function of [itex]A\subset X[/itex] is measurable iff A is measurable. ( Remark: In the text we gave the proof for a Borel space only. )

    P.S. I seem to have figured it out. In the book, for a general measurable space, a measurable function f is defined in such a way that the set on which f is 0 is eliminated. Since [itex]\chi_A(A^c)=0, A^c[/itex] need not be measurable. Thank you both!
     
    Last edited: May 25, 2012
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