I have a question regarding the application of the Fubini's Theorem to the expectation of the product of two random variables.
Let $X,Y$ be two random variables defined on the probability space $(\Omega, \mathcal{F},\mathbb{P})$, $X:\Omega\rightarrow \mathbb{R}$, $Y:\Omega\rightarrow \mathbb{R}$. Assume $E(XY)$ exists.
(case 1): Suppose $X,Y$ continuous with pdf $f^{X,Y}$ and $\int_{\mathbb{R}\times\mathbb{R}} |xy\text{ } f^{X,Y}(x,y)|\text{ }d(x,y)<\infty$. Then $$ E(XY):=\int_{\mathbb{R}\times\mathbb{R}} xy\text{ } f^{X,Y}(x,y)\text{ }d(x,y)\underbrace{=}_{\text{Fubini}}\int_{\mathbb{R}}\int_{\mathbb{R}}xy\text{ } f^{X,Y}(x,y)\text{ }dx dy $$
Question: can we generalise case 1 to the case in which $X,Y$ could be both discrete, both continuous or mixed?
My attempt: Let $P^{X,Y}$ be the probability distribution induced by $X,Y$ on the probability space $(\mathbb{R}^2,\mathcal{B}(\mathbb{R}^2))$, where $\mathcal{B}(\mathbb{R}^2)$ is the Borel $\sigma$-algebra on $\mathbb{R}^2$ . In other words, $P^{X,Y}$ is the measure on the product space $(\mathbb{R}^2,\mathcal{B}(\mathbb{R}^2))$. Then, $$ E(XY):=\int_{\mathbb{R}\times\mathbb{R}} xy \text{ }dP^{X,Y}(x,y) $$
Assume $\int_{\mathbb{R}\times\mathbb{R}} |xy| \text{ }dP^{X,Y}(x,y)<\infty$. At this point how can I apply the Fubini's Theorem? Is $P^{X,Y}(x,y)$ a product measure on $(\mathbb{R}^2,\mathcal{B}(\mathbb{R}^2))$ My intuition is that $P^{X,Y}(x,y)$ is a product measure only if $X$ and $Y$ are independent. Maybe I could use the fact that $P^{X,Y}=P^{X|Y}\times P^Y$? I'm confused on this point and any hint would be really appreciated.