Independence of a random variable $X$ from itself In our lecture on probability, my professor made the comment that "a random variable X is not independent from itself." (Here he was specifically talking about discrete random variables.) I asked him why that was true. (My intuition for two counterexamples are $X \equiv 0$ and $X$ s.t. $$m_X(x) = \begin{cases}1, &\text{ if } x = x_0\\ 0, &\text{ if }x \neq x_0.)\end{cases}$$
In these cases, it seems that $\mathbb{P}(X \leq x_1 , X \leq x_2) = \mathbb{P}(X \leq x_1) \cdot \mathbb{P}(X \leq x_2)$.
My professor's response was, "The independence from or dependence of $X$ on itself depends on the definition of the joint distribution function $m_{X,X}$, which is essentially arbitrary."
Can someone help me to understand this?

The only events that are independent of themselves are those with probability either $0$ or $1$.  That follows from the fact that a number is its own square if and only if it's either $0$ or $1$.  The only way a random variable $X$ can be independent of itself is if for every measurable set $A$, either $\Pr(X\in A)=1$ or $\Pr(X\in A)=0$.  That happens if and only if $X$ is essentially constant, meaning there is some value $x_0$ such that $\Pr(X=x_0)=1$.
