why is this $\bar X$ a.s. irrational? Suppose that $X_1, X_2,\cdots, X_n$ are iid random variables from $N(\theta,1)$, $\theta$ is rational. Then we know that $\bar X \sim N(\theta,1/n)$. It is said that $\bar X$ is almost surely irrational. I am wondering why is it irrational a.s.? How can I interpret it?

Let me denote $\bar X$ by $Z$. Since $Z$ is a random variable with continuous density, the probability of the event $E_q$ that $Z = q$ is $0$ for any fixed real $q$. Now, the event $Z \in \mathbb Q$ can be written as $\bigcup_{q \in \mathbb Q} E_q$; hence by countable additivity, this event also has a zero probability. 

We effectively ignored a lot of the details given in the question (e.g., $Z$ is normally distributed, that $\theta$ is rational). The conclusion does not change as long as $Z$ has a continuous distribution. 
