Let $\Omega = \{3,4\}$ be a sample space. Define a probability measure $P$ on $\Omega$ by $P[3] = 0$ and $P[4] = 1$.
Let $X$ be the random variable defined by $X(\omega) = \omega$.
Definition: A random variable $X$ is degenerate if there is some real number $b$ such that $P[X = b] = 1$.
I claim that my random variable $X$ above is an example of a degenerate random variable that is not a constant function on all of $\Omega$ since $X(3) \neq X(4)$.
Am I right?
I'm self-studying from an undergraduate intro to probability textbook and this problem asks to come up with such an example. It's from the very first chapter.