Understanding conditional probability 3

Question

The expectation of $x_0$ is $\mu = E(x_0) = m_0$

The expectation of $x_0$ given $\mu$ is $x^f = E[x_0\mid \mu]=\mu$

Can someone explain how the $E[x_0\mid \mu] =\mu$? I am not able to understand how this is written?

Answer 1

$\mu$ is a constant value.   The expectation of a random variable, given the value of some constant is just the expectation of the random variable.   It just happens in this case that the expectation evaluates to that constant.

There are many other cases.   Let $c$ be the speed of light, a constant value.   $\mathsf E(x_0\mid c) = \mathsf E(x_0) = \mu$