What is the sample space size for a set like ${(x,y)∈[0,1]^2}$? Should it be infinite? Since there are infinite members of this set so the cardinality is uncountably infinite. Similarly, what is the sample space size for the unit interval sample space $∈ [0,1]$?
Is there a definition for defining this kind of sample space size?
The reason I am asking for this is calculating $P=|A|/|$sample space$|$, $A$ being event space. if sample space$=\\{(x,y)∈[0,1]^2|x ≥ y\\}$, $A=\\{(x,y)∈[0,1]^2| x ≥ 1/2,y ≤ 1/2,x ≤ y + 1/2\\}$, then $P=1/4$ but since both are infinite we just calculate the areas of the two spaces, how do we know the event space and the sample space have the same density on average (occupied by $(x,y)$ points)?
Let it be that $X,Y$ are random variables that have a joint probability uniform on $\Omega:=\{\langle x,y\rangle\in[0,1]^2\mid x\geq y\}$.
Uniformity is characterized by a constant PDF $f$ on $\Omega$ that satisfies: $$\iint_{\Omega}f(x,y)dxdy=\mathsf P(\Omega)=1$$Substituting $f(x,y)=c$ in this we find that $\frac12c=1$ or equivalently that $c=2$
So the distribution is determined by PDF $f:\mathbb R^2\to\mathbb R$ prescribed by $\langle x,y\rangle\mapsto2$ if $\langle x,y\rangle\in\Omega$ and $\langle x,y\rangle\mapsto0$ otherwise.
Then for any event $A\subseteq\Omega$ we find:$$\mathsf P(A)=2\iint_Adxdy$$
This can be applied on $A=\{\langle x,y\rangle\in[0,1]^2\mid x\geq 1/2\text{ and }y\leq 1/2\text{ and }x\leq y + 1/2\}\subseteq\Omega$.