show that $\Delta \in \mathcal{B} \otimes \mathcal{P}$

Question

let $\lambda$ be the Lebesgue measure on the sigma algebra $ \mathcal{B} = \mathcal{B} ([0,1])$ and let $\mu$ be the counting measure on the sigma algebra $ \mathcal{P} = \mathcal{P}([0,1])$

let $\Delta = \{(x,x)\, \mid \, x \in [0,1] \}$

show that $\Delta \in \mathcal{B} \otimes \mathcal{P}$

one way to prove it would be to write it as $\Delta_1 \times \Delta_2$ such that $\Delta_1 \in \mathcal{B} $ and $\Delta_2 \in \mathcal{P} $

which I couldn't do

my attempt :

$\Delta = \bigcup_{x \in [0,1]} \{(x,x)\} $

and $\{(x,x)\} = \{x\} \times \{x \} \in \mathcal{B} \times \mathcal{P} $

now if the union was countable I would be done but it's not.

hints or any kind help are very welcome.

Answer 1

Can you verify that $\Delta^{c}=(\cup_q [(y>q\cap (x<q)]) \cup (\cup_q [(x>q\cap (y<q)])$ where $q$ ranges over rational numbers and conclude that $\Delta^{c}$, hence $\Delta$ belongs to $\mathcal B \otimes \mathcal P$?. [$\Delta^{c}$ is the complement of $\Delta $]