Measurability of a subset of the diagonal

Question

Let $(X, \mathcal{A},\mu)$ be a $\sigma$-finite measure space such that the diagonal $\{(x,x): x \in X\}$ is measurable in the measure space $(X \times X, \mathcal{A} \otimes \mathcal{A},\mu \otimes \mu)$.

Let $A$ be a measurable subset of $X$.

1. Is $\{(x,x): x \in A\}$ measurable ?

2. If it is false, under which condition on the measure space $X$ it is true ?

Answer 1

$A\times X$ is measurable.

Take the intersection of that with the diagonal.