2
$\begingroup$

The function defined by $f:X{\rightarrow}Y$ is continuous. If $X$ is endowed with discrete metric. 

Then, ball $B(x,\dfrac{1}{2})=\{x\}$ for all $x\in X $.

For any $\epsilon >0$

$f(B(x,\dfrac{1}{2}) \subset B(f(x), \epsilon) $

Now,consider characteristic function

$\chi_{A}:X{\rightarrow}R$ defined by

$\mathcal{X}_E(x) =\begin{cases}1, if& x \in E\\ 0, & if& x \not\in E \\ \end{cases}$

is discontinuous if $X=R$ endowed with the standard metric. 

But, is continuous if $X$ is $R$ endowed with the discrete metric. 

And Now,

let (X, M) is a measurable space ,

where sigma algebra $M =\{\phi, X, \{a\}, \{b\}\}$ and $X=\{a, b\}$.

Then , the characteristic function is measurable.

Let $E\subset R$,

$\mathcal{X}_A^{-1}(E) =\begin{cases}X, & 0,1 \in E \\A, & 1 \in E, 0 \notin E \\A^C, & 1 \notin E, 0 \in E \\\emptyset, & o.w.\end{cases}$

This gives our characteristic function is measurable and continuous.  And according to definition of simple function a complex measurable function s on a measurable space $X$ whose range is only finitely many points is called simple function.  Hence,the characteristic function consider above is continuous simple function. 

Thus,giving us a continuous simple function. I am correct or wrong please check and correct my understanding .I have used Rudin book of Real and complex analysis and some online notes .

$\endgroup$
2
  • 2
    $\begingroup$ Did you mean $X$ has the discrete metric? If so, then every function from $X$ to any topological space is continuous. $\endgroup$ – Chris Eagle Dec 3 '20 at 22:35
  • 3
    $\begingroup$ Yes, I mean X is discrete metric. What about the function being measurable. $\endgroup$ – SHREYA PANDEY Dec 4 '20 at 0:47
0
$\begingroup$

If a topological space $X$ is discrete then any its subset is open (and so Borel) so any function from $X$ to a topological space is continuous, and any function from $X$ to a measurable space is Borel-measurable.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.