What is the meaning of a distribution measure? On the set $\mathbb{R}$ of all real numbers, I then randomly pick up a number $x$. Assume that $\mathbb{R}$ is unevenly distributed, namely, for example, let's say, it is more likely to get a big, positive number than a negative one; or it's more likely to pick up an irrational number than a rational one.
To fomulate this, one way is to consider a $\sigma$-algebra on $\mathbb{R}$ (for example, let's consider the Borel $\sigma$-algebra) and a probability measure $\mathbb{P}$ on it. That is, we consider a probability space $(\mathbb{R},\mathcal{B}(\mathbb{R}),\mathbb{P})$. This formulation is very clear to me.
Now, let's consider another way to formulate our situation. To say that  $x$ is randomly chosen, I consider it as a random variable. That is: let $(\Omega,\mathcal{F},\mu)$ be another probability space and let $X:(\Omega,\mathcal{F},\mu) \rightarrow (\mathbb{R},\mathcal{B}(\mathbb{R}))$ be a random variable. So, now I imagine that when I randomly pick a $x$ then I can think of it as $X(\omega)$, for some $\omega\in\Omega$. And, how to measure the likelihood of an event! I consider the distribution measure $P_{X}$, defined by
$$P_{X}(A)=\mu(\{\omega\in\Omega:X(\omega)\in A\}),\quad \text{for any }A\in\mathcal{B}(\mathbb{R}).$$
My question: Are the two formulations identical?
A similar situation is as follows:
I randomly pick up a real number $x$ (as in the above experiment) and my friend randomly picks up a number in an infinite sequence containing only 0 and 1 (some thing like: 0,1,1,0,0,1,0,...). Since the distributions are uneven, one way to formulate the problem is to consider the probabilty space $(\mathbb{R}\times Y,\mathcal{B}(\mathbb{R})\otimes \mathcal{Y},\mathbb{P})$. Here $Y=\{0,1\}$ and $\mathcal{Y}$ is the $\sigma$-algebra $\{\{0\},\{1\},\{0,1\},\emptyset\}$. And $\mathbb{P}$ is some probability measure to quantify how likely $(x,y)$ happens when I randomly pick up $x$ and my friend randomly chooses $y$.
But, is the following formulation the same? Again, I consider two random variables: $X:(\Omega,\mathcal{F},\mu)\rightarrow (\mathbb{R},\mathcal{B}(\mathbb{R}))$ and $\mathbb{Y}: (\Omega,\mathcal{F},\mu)\rightarrow (Y,\mathcal{Y})$. Here the domains of these random variables are the same. To measure the likelihood of an event, I consider the distribution measure $P_{(X,\mathbb{Y})}$ defined by:
$$P_{(X,\mathbb{Y})}(A,B)=\mathbb{P}(\{\omega\in\Omega:X(\omega)\in A, \mathbb{Y}(\omega)\in B\}),\quad \text{for any }A\in \mathcal{B}(\mathbb{R}), B\in\mathcal{Y}.$$
 A: A measurable function (also know as random variable, or observable) $X$ from a probability measure space $(\Omega,\mathscr{F},\mathbb{P})$ into some measurable space space $(S,\mathscr{S})$, induces a probability measure $P_X$ on $(S,\mathscr{S})$ given as
$$ P_X(A)=\mathbb{P}[\{X\in A\}],\quad A\in\mathscr{S}
$$
This probability measure $P_X$ is called the distribution of $X$. This gives the mathematical framework to measure the "likelihood" that our observables fall within some range $A$ of values.
Given two random variables $X:(\Omega,\mathscr{F},\mathbb{P})\rightarrow (S,\mathscr{S})$ and $Y:(\Omega,\mathscr{F},\mathbb{P})\rightarrow (T,\mathscr{T})$, the function $U=(X,Y):(\Omega,\mathscr{F},\mathbb{P})\rightarrow(S\times T,\mathscr{S}\otimes\mathscr{T})$ is also a random variable (Here $\mathscr{S}\otimes\mathscr{T}$ is the $\sigma$--algebra generated but the sets of the form $A\times B$ where $A\in\mathscr{S}$ and $B\in\mathscr{T}$. The joint distribution of $X$ and $Y$, $P_{X,Y}$ is the distribution of $U$. So is $A\times B\in\mathscr{S}\otimes\mathscr{T}$,
$$
P_U(A\times B):=P_{X,Y}(A\times B)=\mathbb{P}[\{X\in A\}\cap \{Y\in B\}]
$$
