Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ a continuous and injective map. Let $A \subset \mathbb{R}^n$ a Borel set, i.e. $A \in \mathscr{B}(\mathbb{R}^n)$, where $\mathscr{B}(\mathbb{R}^n)$ denote the Borel $\sigma$- algebra. Show that the image of $A$ is also a Borel set, i.e. $f(A) \in \mathscr{B}(\mathbb{R}^n)$.
My attempt:
Let $A \subset \mathbb{R}^n$ a Borel set. Since $f$ is injective, then we have that $A = f^{-1}(f(A))$. NTS: $f(A)$ Borel set. We suppose by contradiction that $f(A)$ is not a Borel set, so $f(A)$ is not an open or a closed set.
I'd like to find a contradiction using the continuity of $f$ but I don't see how.
Any suggestions? Thanks!
Best Answer
We want to show that $$\mathscr{B} (\mathbb{R}^n) \subset \mathscr{A}:= \{A \subset \mathbb{R}^n : f(A) \in \mathscr{B}(\mathbb{R}^n) \}$$
By the definition of Borel $\sigma$-algebra it is enough to show that $\mathscr{A}$ is a $\sigma$-algebra and it contains all open sets.
Proof: Indeed $f(\emptyset)=\emptyset$ and so $\emptyset\in\mathscr{A}$. Also write $\mathbb{R}^n = \bigcup_{m=1}^{\infty} I_m$ where $I_m$ is the $n$-dimensional cube with edge length $m$, this is a compact subset and so $f(\mathbb{R}^n) = \bigcup_{m=1}^\infty f(I_m)$ now $f(I_m)$ is compact hence closed and so is Borel measurable, we cocnlude that the union is Borel measurable and so $\mathbb{R}^n\in\mathscr{A}$.
Now if $A\in\mathscr{A}$ then $f(\mathbb{R}^n\backslash A) = f(\mathbb{R}^n)\backslash f(A)$ (by injectivity) and so $\mathbb{R}^n\backslash A \in \mathscr{A}$.
Finally if $A_1,A_2,...\in\mathscr{A}$ then $f(\bigcup A_i ) =\bigcup_i f(A_i)$ and so $\bigcup A_i \in \mathscr{A}$. We conclude that $\mathscr{A}$ is a $\sigma$-algebra.
Proof: This is not that hard, look at here Every open set in $\mathbb{R}^n$ is the increasing union of compact sets. Every open set is a countable union of compact sets and so you can argue as we did with $\mathbb{R}^n$.
Thus by Claims 1 and 2 we have that $\mathcal{A}$ is a $\sigma$-algebra and it contains all open sets. Since $\mathcal{B}(\mathbb{R}^n)$ is the minimal with such property, we have the desired inclusion.