This is an exercise in Folland's real analysis text.
Consider $X = (I,\mathcal{B}_I, \mathcal{L})$ and $Y = (I,\mathcal{B}_I,\mathcal{C})$, where $\mathcal{B}_I$ is the Borel $\sigma$-algebra on $I$, and $\mathcal{L}$ and $ \mathcal{C}$ are the Lebesgue and counting measures, respectively. Last, let $D = \{(x,x): x \in I\}$.
I wish to compute:
$$\iint \chi_D d\mathcal{L}d\mathcal{C}$$
$$\iint \chi_D d\mathcal{C}d\mathcal{L}$$
$$\iint\chi_D d(\mathcal{L}\times\mathcal{C})$$
To show that $\sigma$-finiteness is necessary for the Fubini-Tonelli Theorem to hold.
I computed the first two, but I am having trouble computing $(\mathcal{L} \times \mathcal{C})(D)$. I can't approximate $D$ from below as a limit of "rectangles" (i.e., points), since it's uncountable. When I try to approximate it from above, I can't apply continuity of measure, since for any set $R$ of rectangles containing $D$, $(\mathcal{L} \times \mathcal{C})(R) = \infty$, so the conditions for
$$(\mathcal{L} \times \mathcal{C})\left(\bigcap E_i\right) = \lim(\mathcal{L} \times \mathcal{C})(E_i)$$
do not hold. Indeed, it holds that $$\mathcal{C} \left(\bigcap_{i \in \mathbb{N}}(-1+1/i,1-1/i)\right) = 1 \neq \infty = \lim_{i \to \infty}\mathcal{C}((-1+1/i,1-1/i)), $$ so I'm suspicious of this kind of argument in the two-dimensional case.
I could see it being both measure $0$ and measure $\infty$. I see there are other versions of this post, but none of them answer this question in a satisfactory manner. In particular, the two answers I have seen involve outer measures, and hand-waving about cardinality. The first does not answer the question, and the second is too vague.
Best Answer
$\newcommand{\l}{\mathcal{L}} \newcommand{\n}{\mathcal{C}}$ I assume here that $\mathcal{I} = [0,1]$. Using the definition of the product measure: $$(\l \times \n)(D) = \inf \left\lbrace \sum_{n=1}^\infty \l (A_n) \n (B_n) \, : \, (A_n\times B_n)_{n=1}^\infty \text{ are rectangles covering }D \right\rbrace$$
Note that if $(A_n \times B_n)_{n=1}^\infty$ covers $D$, then $(A_n \cap B_n)_{n=1}^\infty$ covers $[0,1]$, so that $\l (A_n \cap B_n) >0$ for some $n\in \mathbb{Z}^+$. But this particular set cannot be finite (otherwise it would have Lebesgue measure zero), so we have $\l (A_n) > 0$ and $\n (B_n) = \infty$. Thus, $$\sum_{n=1}^\infty \l (A_n) \n (B_n) = \infty$$ for any cover of $D$ by rectangles, giving you $(\l \times \n )(D) = \infty$.