Measurability of a function with finite codomain

[hwangii]

Hi all,
I have a simple question as follows:
$f$ is a function from $X$ to $Y$ where
$X=[0,1]$; and
$Y$ is finite, i.e. $\vert Y\vert <\infty$
then is $f$ Borel measurable?
Thank you for your help in advance.

 

[lavinia]

Hi all,
I have a simple question as follows:
$f$ is a function from $X$ to $Y$ where
$X=[0,1]$; and
$Y$ is finite, i.e. $\vert Y\vert <\infty$
then is $f$ Borel measurable?
Thank you for your help in advance.

Why would you think that?

 

[mathman]

Let f be the characteristic function of a non-measurable subset of the unit interval. f is not Borel measurable.