Why the answer in the linked question is treating the integral as Riemann integral and not Lebesgue ones?

Question

Here is the question:

An example of $\int_{X \times Y} f d(\mu \times \lambda )$ is finite but such that $\int_{Y} f_{x} d\lambda $ is infinite for some $x \in (0,1).$

**My questions are: ** 1-If I understand correctly, Why the answer in the linked question is treating the integral as Riemann integral and not Lebesgue ones?

2-Also, how, in Lebesgue integration, we overcomed that at $0$ the $y$ in the denominator makes the integrand undefined?