Question

(8) Give an example of a metric space (X, d), a o-algebra A in X, and a continuous function f X ->R which is not measurable with respect to A. (Does this question make sense if (X, A) is a measurable space without a metric?)

Answer 1

Consider the real line $\mathbb R$ with the usual metric and $\sigma$-algebra $\mathcal A$ consist of only $\mathbb R$ and the empty set $\phi$, that is the smallest $\sigma$-algebra on $\mathbb R$. Consider the identity function $i:(\mathbb R, d,\mathcal A)\to(\mathbb R, d,\mathcal B)$ , where $\mathcal B$ is the Borel sigma algebra.

Then note that is continuous as inverse of any open set $U\subset \mathbb R$ , is which is open in $\mathbb R$(domain).

But note that is not measurable as consider any proper open set $U\subset \mathbb R$ , which is Borel measurable, but note that $i^{-1}(U)=U$ is not in $\mathcal A$.

Note that that the problem does make sense as long as is a topological space, as continuity  only make sense for topological spaces. For a non topological space the problem does not make any sense.

Feel free to comment if you have any doubts. Cheers!