Probability on product spaces I am having some trouble, more of an argument with someone else, about a simple question regarding product spaces. 
Let $X_1,X_2,\dots,X_n$ a set of independent and identically distributed random variables from a population $P\in\mathcal{P}$, where $\mathcal{P}$ is a family of probability measures (non explicitly parameterized). The generic random variable $X$ is a measurable function from a probability space $\left(\Omega,\mathcal{F},P'\right)$ to $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. 
In order to build the random vector containing the elements of the sample, $\tilde{X}=[X_i]_{n\times 1}$, and be able to calculate probability measures of events like $\{X_1<X_2\}$ or $\{X_1=X_2\}$, I decided to build a product space $(\Omega^n=\Omega\times\Omega\dots\times\Omega,\sigma(\mathcal{F}^ n)=\sigma(\mathcal{F}\times\mathcal{F} \dots,\times\mathcal{F}),P=P'\times P'\times\dots\times P')$, and let the vector function $\tilde{X}$ go from this product space to $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$. 
Therefore, the events 


*

*$\{X_1=X_2\}=\{(w_1,w_2,\dots,w_n)\in\Omega^n:X_1(\omega_1)=X_2(\omega_2)\}$
and 

*$\{X_1<X_2\}=\{(w_1,w_2,\dots,w_n)\in\Omega^n:X_1(\omega_1)<X_2(\omega_2)\}$


have a clear meaning. 
However, the person I am having the argument with argues that 


*

*$\tilde{X}$ goes from the original probability space $\left(\Omega,\mathcal{F},P'\right)$ to $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$, and 

*If I keep doing it the way of product spaces it is a lot harder to calculate the probabilities I want. 


If my friend's argument is true, I am having trouble finding the meaning of, and calculating the probabilities of the events above. 
What am I missing?
Best regards, 
JM 
 A: That's a good question. The point is that if you have a random variable 
$$
 X:(\Omega,\mathscr F,P)\to (\Bbb R,\mathscr B(\Bbb R))
$$
and you need to consider, say, two copies of this variable $X_1$ and $X_2$, a direct way would be to define the vector $\tilde X = (X_1,X_2)$ on a product space $(\Omega,\mathscr F,P)\otimes (\Omega,\mathscr F,P)$. This is intuitive, this way always works, and IMHO it's easier to compute the probabilities you've mentioned over a product space - you have clear image of the diagonal in your mind when dealing with $\{X_1 = X_2\}$ and of a subdiagonal triangle when dealing with $\{X_2\leq X_1\}$. The latter make easier computations of the correspondent double integrals.
On the other hand, formally speaking, you don't have to construct a product space in most of the practial cases. That is, most of the probability spaces we're dealing with are standard. For example, since $X$ is a real-valued random variable, you can always take
$$
  (\Omega,\mathscr F,P) = ([0,1],\mathscr B([0,1]),\lambda)
$$
where $\lambda$ is the Lebesgue measure. As a result, the product space is isomorphic to the original space and hence any random vector defined over the product space can be defined over the original space. However, I wouldn't suggest going that way due to the following reasons:


*

*It does not always work: if $\Omega$ has just two elements and $\mathscr F$ is its powerset, then you can't defined $\tilde X$ over the original space.

*I disagree with your friend that it is easier to compute probabilities when defining $\tilde X$ over the original state space, rather than over the product space. 

*It is less intuitive, more technically involved and unnecessary. 
Please, tell me whether the answer is clear to you.
A: You're on the right track. The sample space you want is the product probability space $\prod_1^n (\mathbb{R}, \mu_n)$ where $\mu_n$ is the push-forward measure on $\mathbb{R}$ given by $X_n$. The random variable $X_m$ is then the $m$-th coordinate projection from  $\prod_1^n (\mathbb{R}, \mu_n)$.  
The generalization of this construction to arbitrary family of random variables is due to Kolmogorov, I believe.
