suppose we have a [tex] X = [0,1] [/tex] and a function [tex] f\colon X \to \Re [/tex] where(adsbygoogle = window.adsbygoogle || []).push({});

[tex] f(x) = 1 - |2x -1| [/tex].

i'm bit confused on finding the sigma-algebra generated by this function. This is what i did

[tex] f(x)= \begin{cases}

2 -2x & x \in [\frac{1}{2},1] , \\

2x& x \in [0, \frac{1}{2})

\end{cases}

[/tex]

so then is the sigma-algebra [tex] \sigma(f(x)) = \mathcal{B}([\frac{1}{2},1] \bigcup \mathcal{B}([0, \frac{1}{2}) = \mathcal{B}([0,1]) [/tex] ?

some thing about this doesnt feel quite right to me, could someone show me where i have made a mistake.

Also what is a systematic way or method of finding the sigma-algebra generated by a function.

the i do it is find the pre-image of the function of any open set in [tex] \Re [/tex] it far to easy for me to make mistakes when doing it this way. are alternative methods ?

any comments, help much appreciated

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# Sigma-algebra generated by a function

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