0
$\begingroup$

I want to know if my function $f$ is Borel measurable function or not.

To be clear, I use the terms which are introduced in $\\$https://dspace.mit.edu/bitstream/handle/1721.1/14254/22712180-MIT.pdf?sequence=2 , page 17. $\\$ It says, for any topological space $X$, the Borel sigma-algebra of $X$ is denoted by $\mathcal{B}_X$; it is the smallest sigma-algebra which contains all of the open sets in X. A subset of X is Borel measurable if it belongs to $\mathcal{B}_X$... . . . In particular, a real-valued function $f:X\to \pmb{R}$ is Borel measurable if $f^{-1}(B) \in \mathcal{B}_X$ for all Borel measurable $B \subset \pmb{R}$. The space of all bounded Borel measurable functions on $X$ is denoted by $\mathcal{B}(X)$. $\\$

Now, the function $f$ is $$ f:\pmb{I} \to \pmb{O} $$ where $I=([0,a_1],[0,\infty),[0,\infty),[0,a_2])$ which means that the input space of the function $f$ has four dimension and the ranges of each elements are $[0,a_1],[0,\infty),[0,\infty),[0,a_2]$ respectively($a_1$ and $a_2$ are positive real numbers). The $O$ has only one dimension, $O=[0,o_1]$ ($o_1$ is positive real number). $\\$

I think the input space $I$ is a Borel measurable subset of $\pmb{R}^4$ because every interval is a Borel set.

However, I do not understand why the author used subset notation for $B\subset\pmb{R}$. I think $f^{-1}$ is one of the function which should take the scalar value as input in here. But $B\subset\pmb{R}$ indicates that $B$ is set. So I am not sure how to check the function $f$ is Borel measurable function or not, i.e., want to know $f \in \mathcal{B}(I)$. $\\$

$\endgroup$
  • $\begingroup$ Note that we usually denote $I= [0,a_1]\times [0,\infty)\times [0,\infty)\times [0,a_2)$ also the word "the" in the sentence "Now, the function $f$ is $f:I\rightarrow O$" makes no sense because there are infinitely many functions with such domain\range. $\endgroup$ – Yanko Mar 16 at 20:33
1
$\begingroup$

You haven't told us what the function is, only what its domain and range are, so we can't answer this.

As for the notation $f^{-1}(B)$, it denotes the preimage of the set B, i.e the set of points $x$ for which $f(x) \in B$. A function is measurable if the preimage of each measurable set is measurable.

$\endgroup$
  • $\begingroup$ (+1). If the OP asks whether any function $f:I\rightarrow O$ is Borel measurable then the answer is NO. $\endgroup$ – Yanko Mar 16 at 20:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.