Let $X$ be a set, $F$ a $\sigma$-field of subsets of $X$, and $\mu$ a measure on $F$. A map $F=(f_1,\ldots,f_n)$ of $X$ into $\mathbb{R}^n$ is said to be measurable if each of the coordinate functions, $f_i$, is measurable. Show that $f$ is measurable if and only if $f^{-1}(B)\in F$ for every Borel set $B\subseteq \mathbb{R}^n$.
The function $f_i$ being measurable means that $f_i^{-1}(B)\in F$ for every Borel set $B\in\mathbb{R}$. Therefore, we must show that this is true for every $i=1,\ldots,n$ if and only if $f^{-1}(B)\in F$ for every Borel set $B\subseteq \mathbb{R}^n$.
Borel sets are the smallest $\sigma$-ring containing the open sets. I can't see how to relate Borel sets in $\mathbb{R}$ and Borel sets in $\mathbb{R}^n$.