Consider $\Omega_1 = \mathbb{N}_0$, $\Omega_2 = \mathbb{R}$, $\mathbb{P}_1 = Poi(5)$ and $\mathbb{P}_2 = Normal(1,2)$. On $\Omega = \Omega_1 \times \Omega_2$ we use the the product measure $\mathbb{P}$.
I want to find out if $A = \{\ 1,3,5 \} \times \mathbb{R}$ and $B = \mathbb{N}_0 \times [1, \infty)$ are independent.
I use the rule of product measure $\mathbb{P}(A_1 \times ... \times A_n) = \mathbb{P}_1(A_1) \cdot ... \cdot \mathbb{P}_n(A_n)$ and compute: $$\mathbb{P}(A) = \mathbb{P}(\{\ 1,3,5 \} \times \mathbb{R}) = \mathbb{P}_1(\{\ 1,3,5 \}) \cdot \mathbb{P}_2(\{\ \mathbb{R} \}) = \sum_{k \in \{\ 1,3,5 \}} \frac{5}{k!}e^{-5} \cdot 1 \\ \mathbb{P}(B) = \mathbb{P}(\mathbb{N}_0 \times [1, \infty)) = \mathbb{P}_1(\mathbb{N}_0) \cdot \mathbb{P}_2([1, \infty)) = \sum_{k \in \mathbb{N}_0} \frac{5}{k!}e^{-5} \cdot \frac{1}{2}$$
How do I now compute $\mathbb{P}(A \cap B)$ to check for independence?
Note that $A\cap B = \{1,3,5\} \times [1,\infty )$. Thus by definition of product measure you have $$\Bbb P (A\cap B) = \Bbb P ( \{1,3,5\} \times [1,\infty ) ) = \Bbb P ( \{1,3,5\} ) \Bbb P ( [1,\infty ) ) = \Bbb P (A) \Bbb P (B)$$ as you already calculated the last equality.