Existence of two independent random variables on a finite sample space. The sample space $\Omega$ contains five elements with positive probability. Are there two independent random variables on $\Omega$, each of which takes five different values?



My solution is the following. Suppose there are random variables $X$ and $Y$, each of which takes five different values. Let $\omega_0$ be one of the outcomes and $X(\omega_0) = x_0, Y(\omega_0) = y_0 $ and $\forall i X(\omega_i) \neq x_0, Y(\omega_i) \neq y_0 $. Then $P(x = x_0, y = y_0) = p $ and $P(x = x_0) \cdot P(y= y_0) = p^2 $. Therefore the independence criteria is not satisfied and the answer to question is no. 

Is that a correct solution?
It suffices to assume that there exists $\omega\in \Omega$ with $0&lt;\mathsf{P}(\{\omega\})&lt;1$, where $\mathsf{P}$ is a probability measure on $(\Omega,2^{\Omega})$ because for any injective functions $X,Y:\Omega\to\mathbb{R}$ (random variables taking distinct values for each $\omega$), $\sigma(X)=\sigma(Y)=2^{\Omega}$.
I think, as I wrote in the comment, that the proposed solution from the OP, after the changes he made, is correct. Here I propose my approach, which is based on conditional probabilities,  that maybe shows a different point of view. I try to be a bit heavy with the notation to make the argument clearer.

We can consider $\Omega=\{1,2,3,4,5\}$, with probabilities $p_1,..p_5$, as sample space. 

Let $X$,$Y$ two generic random variables, so that $X(i)=x_i$ and $Y(i)=y_i$. We observe that we can define $Z=(X,Y)$ so that:

$Z(i)=(x_i,y_i)$

It is clear than that:

$p(\{X=x_i\} \cap \{Y=y_k\})=p(Z=(x_i,y_k))=\delta_{i,k}p_i$ [1],

because if $i$ is not equal to $k$, $(x_i,y_k)$ does not belong to the codomain of $Z$. If $i$ is equal $k$ than the event $\{X=x_i\}$ happens only for $\omega \in \Omega=i$, for which $Y$ will be automatically equal to $y_i$. Thus the probability in [1] is simply $p_i$. 

Using [1], $p(X=x_i|Y=y_k)=\frac{p(\{X=x_i\} \cap \{Y=y_k\})}{p(Y=y_k)}=\delta_{i,k}$ [2]

, where $\delta$ is the Kronecker symbol. If the two variables were independent [1] could be factored and we would obtain $p(X=x_i|Y=y_k)=p_i$.

Therefore the two variables cannot be independent. In fact given the value of $Y$ one can reconstruct in a unique way the value of X and viceversa (this is the interpretation of [2]: when conditioned on the value of $Y$, $X$ becomes deterministic). Told another way the value of X is highly "correlated" with the value of Y.