Maybe this is just a stupid question, but I have tried very hard and get no luck. Here is the question:


Let $S\_{k-1}$ be the unit sphere in $R^k$, i.e., the set of all $u\in R^k$ whose distance from the origin 0 is 1. So it is true that every $x\in R^k$, except for $x=0$, has a unique representation of the form $x=ru$, where $r$ is a positive real number and $u\in S\_{k-1}$. Thus $R^k- \lbrace 0\rbrace$ may be regarded as the cartesian product $(0,+\infty)\times S\_{k-1}$. So there is a one-to-one mapping $\phi$ of $R^k$ onto $(0,+\infty)\times S\_{k-1}$.


Let $\mathfrak{B}(X)$ denote the $\sigma$-algebra generated by open sets in $X$. Let $\mathfrak{B}(X)\times \mathfrak{B}(Y)$ denote the $\sigma$-algebra generated by the form $A\times B$, which $A\in \mathfrak{B}(X)$ and $B\in \mathfrak{B}(Y)$.


My problem: if $A\in \mathfrak{B}(R^k-\lbrace 0 \rbrace)$, is it true that $\phi(A) \in \mathfrak{B}((0,+\infty))\times \mathfrak{B}(S\_{k-1})$.


I think this problem is true, trivially. If this problem is just a well-known result, please show me some references. Thank you very much!



Polar Coordinates and Borel sets.
The point here is that $\phi^{-1}:(0,+\infty)\times S\_{k-1}\to\mathbb R^k\setminus\{0\}$
is actually continuous and thus Borel-measurable. Hence $\phi$ maps Borel sets to Borel sets.


