- #1
cappadonza
- 27
- 0
suppose we have a [tex] X = [0,1] [/tex] and a function [tex] f\colon X \to \Re [/tex] where
[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 doesn't 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
[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 doesn't 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