Proving product measures: Homework Being an engineer I'm at loss how to prove the following exercises, and I would appreciate any comments.


*

*Prove Fubini's theorem for an $\mathcal{L}^1$ integrable function $f$. Here's my stab at it.
For a non-negative function $g(x_1,x_2)$ Fubini asserts that $\mu(g) = \mu_2(I^g_2) = \mu_1(I^g_1)$, where $I^g_1$ is the integral of $g$ w.r.t to $x_2$. Building on this result it's sufficient to say that 
$$ f \in \mathcal{L}^1(S,\Sigma,\mu) \implies |f|< \infty$$
and since $|f|= f^+ + f^-$ where $f^+ := \max(f,0)$ and $f^- := \max(-f,0)$, and both are non-negative. By linearity of the integral we have that the results given for non-negative $g$ hold for $\mathcal{L}^1$ integrable $f$ since
$$ \mu(f) = \mu(f^+ - f^-) < \mu(|f|) \equiv \mu(f^+ + f^-) < \infty$$
The integral of $f$ then equals
$$ \mu(f) = \mu_2(I^{f^+}_2 - I^{f^-}_2) = \mu_1(I^{f^+}_1 - I^{f^-}_1) =  \mu_1(I^{f^+}_1) -\mu_1( I^{f^-}_1) $$

*Let $X_1,X_2\;: \Omega\rightarrow \mathbb{R}$ be given, The vector mapping $X = (X_1,X_2)\;:\Omega\rightarrow \mathbb{R}^2$ is a random vector iff $X_i$ are random variables. I've based my ideas on Billingsley 13.2. The random vector is $\mathcal{F}$ measurable if we represent the RV's as
$$X(\omega) = (X_1(\omega),X_2(\omega))$$
and then for each omega
$$[\omega\: : X_1(\omega)\le a_1,X_2(\omega)\le a_2]   = [\omega\: : X_1(\omega)\le a_1]\cap  [\omega\: : X_2(\omega)\le a_2] \in \mathcal{F}.$$
By definition, each intersection is an element of $\mathcal{F}$ since each $X_i$ is measurable.
 A: I think you've got the right idea, but you need to be more explicit in your steps in 1 and haven't finished 2.
For number 1, think about what you actually want to show. The question is asking you show that $\mu(f) = \mu_1(I_1^f) = \mu_2(I_2^f)$. Your method is correct, in that you want to decompose $f$ into $f^+$ and $f^-$, and use the linearity of integrals, then apply Fubini to each of them separately.  Then you want to use the linearity again to recombine them, and get the desired result.
I.e., your start should be something like: 
$$\mu(f) = \mu(f^+ - f^-) = \mu(f^+) - \mu(f^-)$$
Apply Fubini to each of these, and end up with: 
$$\mu(f) = \mu_1(I_1^{f^+}) - \mu_1(I_1^{f^-}) = \mu_1(I_1^{f^+} - I_1^{f^-}) = \mu_1(I_1^{f^+ - f^-}) = \mu_1(I_1^{f})$$
Do the same thing for $\mu_2$. 
For number 2 you have only proved the 'if' direction, and you need to prove the other direction. To complete the proof, assume that $X$ is a random vector and then show that each of the $X_i$ are random variables.
