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Simple question about measurable characteristic function 
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#1
May2412, 04:13 PM

P: 312

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
May2412, 04:47 PM

P: 274




#3
May2412, 06:40 PM

P: 312

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?



#4
May2412, 08:34 PM

P: 350

Simple question about measurable characteristic function
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.



#5
May2412, 09:20 PM

P: 274

Every book I've seen defines a measurable space as a set equipped with a Σalgebra. What book are you using?



#6
May2512, 02:06 PM

P: 312

This is the book I use
http://books.google.com/books/about/...d=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 sigmaring 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 sigmaring: http://en.wikipedia.org/wiki/Sigmaring In the last paragraph: "σrings can be used instead of σfields 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! 


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