Show that if $X$ is a continuous r.v. defined in a probability space, then that probability space is continuous.
My proof:
Justifying by contradiction, suppose $X$ is continuous in a discrete probability space (so that the sample space $\Omega$ is discrete). Then $P(X = x_i) = 0$ for all $x_i$, since for a continuous random variable, the probability of a point is equal to zero. Therefore $X$ wouldn't satisfy the axioms of probability. Therefore, the probability space must be continuous.
I'm not feeling too sure about this "proof", in the sense that someone correcting it might not give me full marks. It is good/rigorous enough?