\(\displaystyle f: R^m -> R^n \) is a Borel function (\(\displaystyle R\) is real numbers).
F:= { A is subset of \(\displaystyle R^n\) so that f^(-1)(A) is Borel set}
I need to show that F is sigma-algebra and that F contains all open subsets of \(\displaystyle R^n\). I am stuck on part where I need to show that if \(\displaystyle A\) belongs to F, then \(\displaystyle A^c\) (complement of \(\displaystyle A\)) belongs to F as well ( (2) sigma-algebra characteristic). It seems that by definition if \(\displaystyle A\) belongs to F, then it's open set and therefore \(\displaystyle A^c\) will not be open and I just can't find a way to show that f^(1)(\(\displaystyle A^c\)) is Borel set.
I would like to see detailed explanation (with use of definitions and such) from whoever solves this problem.
F:= { A is subset of \(\displaystyle R^n\) so that f^(-1)(A) is Borel set}
I need to show that F is sigma-algebra and that F contains all open subsets of \(\displaystyle R^n\). I am stuck on part where I need to show that if \(\displaystyle A\) belongs to F, then \(\displaystyle A^c\) (complement of \(\displaystyle A\)) belongs to F as well ( (2) sigma-algebra characteristic). It seems that by definition if \(\displaystyle A\) belongs to F, then it's open set and therefore \(\displaystyle A^c\) will not be open and I just can't find a way to show that f^(1)(\(\displaystyle A^c\)) is Borel set.
I would like to see detailed explanation (with use of definitions and such) from whoever solves this problem.