# Borel measurable function on the Euclidean space $\pmb{R}^4$

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 $$f \in \mathcal{B}(I)$$. $$\\$$

• 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. – Yanko Mar 16 at 20:33

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