$\textbf{Corollary}$: Let $X,Y$ be locally compact Hausdorff spaces. Let $\mu$ and $\nu$ be regular Borel measures on $X$ and $Y$ respectively. If $E$ is a Borel subset of $X\times Y$ that is included in a rectangle whose sides are Borel sets that are $\sigma$-finite under $\mu$ and $\nu$ respectively, then
a) the functions $x\rightarrow \nu(E_x)$ and $y\rightarrow \mu(E^y)$ are Borel measurable, and
b) $(\mu\times \nu)(E)=\int_X\nu(E_x)\mu(dx)=\int_Y\mu(E^y)\nu(dy)$
The problem I am trying to solve:
Show that the corollary fails if $E$ is an arbitrary Borel subset (or even closed) subset of $X\times Y$. More precisely, show that part (a) can fail and that part (b) can fail even (a) holds. As a hint, let $X$ be $\mathbb{R}$ with the usual topology and Lebesgue measure and let $Y$ be $\mathbb{R}$ with the discrete topology and counting measure.
I read the definition of discrete topology on $\mathbb{R}$, which according to my understanding would be the collection of all singletons and arbitrary unions and intersections of them (which already challenges the fact that $\nu$ is supposed to be regular, since the hint suggests to consider the counting measure, but let's ignore this for a moment). I have not fully grasped what discrete topology means. Then, the usual topology on $\mathbb{R}$ is the one formed by open intervals.
Let $\mu$ be the Lebesgue measure (I will use $\lambda$ instead of $\mu$), and $\nu$ be the counting measure.
Then let $E=V\times \emptyset$, where $V$ is Vitali's set. Since $\nu(\emptyset)=0$, $E$ is a Borel subset of $X\times Y$ but $y\rightarrow \lambda(E^y)=\lambda(V)$ would not be Borel measurable. For $x\rightarrow \nu(E_x)$, I would try to use a similar argument/idea.
I don't really know how to show that (b) can fail even if (a) holds.
So, I would appreciate if someone 1) can help me clarify what discrete topology on $\mathbb{R}$ means, 2) can tell whether my approach to prove that part is valid, and 3) can give me a hint on to show that (b) can fail even if (a) holds.
I found other answers on this website that pretty much answer my question. I'll add them here in case someone needs them in the future.
This does not answer directly but gives insight, This one is very comprehensive, it helped a lot, This one is quick to read.