I have a question related to the definition of expectation of a random vector, in particular, to its relation (if any) with Fubini's Theorem.
Consider the random vector $X:=(X_1,X_2,X_3)$ of dimension $3$ taking values in $\mathbb{R}^3$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Let $x=(x_1,x_2,x_3)$ be a realisation of $X$.
In the following I will list all the definitions of expectation of $X$ that I can think about with some questions. I would like some help for answering those questions and correcting the definitions, if necessary
(1) Suppose $X_1,X_2,X_3$ are continuous random variables. Let $f(x)$ denote their joint probability density function. Then $E(X):=\int_{\mathbb{R}^3} x f(x)d(x_1,x_2,x_3)$ (Riemann integral)
Question: Is this definition of $E(X)$ equivalent to $\int_{\mathbb{R}}\int_{\mathbb{R}} \int_{\mathbb{R}} xf(x)dx_1 dx_2 dx_3$ if $\int_{\mathbb{R}^3} |f(x)|d(x_1,x_2,x_3)<\infty$ (Fubini's theorem for Riemann integral)?
(2) Suppose $X_1,X_2,X_3$ are continuous random variables. Let $F(x)$ denote their joint cumulative distribution function. Then $E(X):=\int_{\mathbb{R}^3} x dF(x)$ (Lebesgue integral)
(3) For $X_1,X_2,X_3$ discrete or continuous, $E(X):=\int_{\Omega} X(\omega) d\mathbb{P}$ (Lebesgue integral)
(4) Suppose $(X_1,X_2,X_3)$ have joint density $\tilde{f}(x)$ with respect to a $\sigma$-finite product measure $\mu\times v\times p$. Then, $E(X):=\int_{\mathbb{R}^3}x \tilde{f}(x)d(\mu,v,p)$ (Lebesgue integral)
Question: The Fubini's theorem tells that, if $x\tilde{f}(x)$ is integrable on $\mathbb{R}^3$ [which implies that $-\infty<E(X)<\infty$, right?] then $E(X):=\int_{\mathbb{R}^3}x \tilde{f}(x)d(\mu,v,p)=\int_{\mathbb{R}}\int_{\mathbb{R}} \int_{\mathbb{R}} x \tilde{f}(x)d\mu dv dp$. Correct?