Show that all functions $X : \Omega \longrightarrow \mathbb{R}$ defined in a discrete probability space is a random variable. $\Omega$ is a sample space
Answer:
Because the definition of a random variable is exactly of a function $X$ such that $X : \Omega \longrightarrow \mathbb{R}$. Besides, given that the function is defined in a probability space, $X$ satisfies the axioms and basic properties of probability.
I think that my answer, while not incorrect, is lacking and not rigorous enough. A Math PhD such as my probability professor might not give me full marks for this answer, so how can I improve it?