Let $\{X_i\}_{i=1}^n$ be a set of iid random variables where $X$ has probability function $f$ with $E[X] = \mu$.
In my statistics book, it defines $E[X_i] = \mu$, and I'm curious why this should be intuitively obvious.
I know that $E[X_i]$ is the theoretical mean of $X_i$, but I don't understand why it should be clear that this expected value is equal to the population mean $\mu$.
On
This is rather trivial and has nothing to do with population means.
You are told that $E[X]=\mu$, that is, $\mu$ is, by definition, the expected value of $X$.
Now, $\{X_i\}_{i=1}^n$ is a set of iid random variables, each of which has the same distribution, that of $X$. Hence to write $E[X]$ is the same as $E[X_i]$, for any given $i$.
The $X_i$ have the have the same cdf as $X$ i.e. $F_{X_{i}}(x)=F_{X}(x)$ for all $x$. In particular the measures induced on $\mathbb{R}$ by $F_{X_{i}}$ and $F_{X}$ (namely, $P_{X_{i}}$ and $P_{X}$) are the same. Hence $$ \mu=EX=\int t\,dP_{X}(t)=\int t\,dP_{X_{i}}(t)=EX_i $$ for all $i$.