A Non-Counterexample to the Fubini Theorem with Counting and Lebesgue Measures 
Let $X = Y = [0,1].$ Let $\mathcal{B}$ denote the Borel $\sigma$-algebra. Let $m$ denote the Lebesgue measure on $[0,1]$, and let $\mu$ denote the counting measure on $[0,1].$ Prove that $D = \{(x,y) : x = y \}$ is measurable with respect to $\mathcal{B} \times \mathcal{B}.$ Furthermore, prove that $$\int_X \int_Y \chi_D(x,y) \, \mu(dy) \, m(dx) \neq \int_Y \int_X \chi_D(x,y) \, m(dx) \, \mu(dy).$$ Explain why this does not contradict the Fubini Theorem.

Given that $D$ is measurable with respect to $\mathcal{B} \times \mathcal{B},$ we believe that we can prove the statement about the integrals. We claim that we have
\begin{align*}
\int_X \int_Y \chi_D(x,y) \, \mu(dy) \, m(dx) &= 1, \text{ but} \\ \\
\int_Y \int_X \chi_D(x,y) \, m(dx) \, \mu(dy) &= \infty;
\end{align*}
however, we are not certain about this, and we cannot prove that $D$ is measurable. We would appreciate any hints or tips on how to approach this problem.
 A: @Diesirae92 explains the measurability and lack of contradiction; we confine ourselves to calculations.
In the first inner integral,
$$
\int_Y \chi_D(x,y) \, \mu(dy)
$$ $\chi_D(x,y) \, \mu(dy)$ basically means, "if you fix an $x\in X$, how many times does the slice at $x$ intersect with $D$?" The answer is clearly once, at a single point, so the counting measure returns $\mu(\{x\})=1$. Then $\int_Y \chi_D(x,y) \, \mu(dy) = 1$, and then $\int_X 1\,m(dx)=1$, yielding
$$
\int_X \int_Y \chi_D(x,y) \, \mu(dy) \, m(dx) = 1
$$
For the second part, the inner integral
$$
\int_X \chi_D(x,y) \, m(dx)
$$asks, "if you fix a $y\in Y$, what is the Lebesgue measure of the intersection with $D$?" The intersection is still a single point, but now this has Lebesgue measure zero. So $\int_X \chi_D(x,y) \, m(dx) =0$, whence $\int _Y 0\,\mu(dy)=0$, and
$$
\int_Y \int_X \chi_D(x,y) \, m(dx) \, \mu(dy) = 0
$$
It is interesting to note that
$$
\int_{X\times Y} \chi_D(x,y) \, d(m\times \mu)= \infty
$$Very roughly, this amounts to asking, "how many points are in $D$," and of course the answer is the cardinality of $\mathbb{R}$.
