Sigma-algebra generated by a function

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In summary, we have a function defined on the interval [0,1] and we are looking to find the sigma-algebra generated by this function. One method to do this is to find the pre-image of any open set in the real numbers. However, this can lead to mistakes. Another method is to realize the symmetry of the function and use this to find the sigma-algebra. This can be done by finding the inverse image for any Borel set, which will result in a union of two intervals in [0,1]. This sigma-algebra will contain "coarser" sets than those in the Borel sigma-algebra on [0,1].
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cappadonza
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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
 
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i think i may have figured it out. i graphed the function [tex] f(x) [/tex] and realized it was symmetrical, [tex] f(x) = f(1-x) \, x \in [0,1][/tex] i then realized to find to generated sigma-field [tex] \sigma(f(x)) = \{ f^{-1}(B) \colon B \in \mathcal{B} \} [/tex] the inverse image for any borel set is the union of two intervals in [0,1] since the function symetrical.
[tex] \sigma(f(x) = \{[\frac{1}{2},1] \bigcap \{1-\frac{B}{2} \colon B \in \mathcal{B} \} \bigcup [0, \frac{1}{2}] \bigcap \{\frac{B}{2} \colon B \in \mathcal{B} \} [/tex]
where [tex] 1-\frac{B}{2} = \{ 1-\frac{x}{2} \colon x \in B\} [/tex]

This seems right to me, since the sigma-algebra contains 'coarser' sets that those contained in [tex] \mathcal{B}([0,1]) [/tex]
 
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1. What is a sigma-algebra generated by a function?

A sigma-algebra generated by a function is a collection of subsets of a given set that is closed under countable unions and complements and contains the sets that are mapped to by the function. In other words, it is the smallest sigma-algebra that contains the pre-images of the function.

2. Why is the concept of sigma-algebra generated by a function important?

The concept of sigma-algebra generated by a function is important in probability theory and measure theory. It allows us to define measurable functions and integration of these functions, which are essential in understanding the behavior of random variables and probability distributions.

3. How is the sigma-algebra generated by a function different from a regular sigma-algebra?

A regular sigma-algebra is defined on a given set, while the sigma-algebra generated by a function is defined on the pre-images of the function. This means that the sigma-algebra generated by a function is a subset of the regular sigma-algebra, but may contain more sets than the regular one.

4. Can any function generate a sigma-algebra?

No, not every function can generate a sigma-algebra. The function must be a measurable function, meaning that the pre-images of the function must be measurable sets. Only then can the sigma-algebra generated by the function be well-defined.

5. How is the sigma-algebra generated by a function used in probability theory?

In probability theory, the sigma-algebra generated by a function is used to define the probability measure on the pre-images of the function. This allows us to calculate the probability of events that are mapped to by the function, which is crucial in understanding the behavior of random variables and probability distributions.

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