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