Given stochastic payoff functions $X\_{1}(t) \dots X\_{K}(t)$, each having a different probability distribution on $[0,1]$, denote the expected value of $X\_i(t)$ by $\mu\_i$, and define $\mu^\* = \max\_{i \in {1\dots K}}\mu\_i$, and define the *pseudoregret* for a certain strategy $I\_t$(which associates an integer between $1$ and $K$ with every $t$) as the following:
$$
\bar{R}\_n = \max\_{i \in {1\dots K}} \mathbb{E}\left[ \sum\_{t=1}^nX\_i(t) - \sum\_{t=1}^n X\_{I\_t}(t) \right]
$$
The paper I'm reading states that this can be written as the following:
$$
\bar{R}\_n = n\mu^\* - \sum\_{t=1}^n \mathbb{E}[\mu\_{I\_t}]
$$
Now I know where the first term comes from, but where does the double expected value come from?(note, in the paper they state that $\mu\_i$ denotes the mean of $X\_i(t)$ but so far as I understand mean and expected value are semantically identical. I may be wrong though.)



Where does the expected value in the restatement of the pseudoregret come from?
Since
$$E(X\_{I\_t}\mid I\_t)=\mu\_{I\_t},$$
by iterated expectation 
$$E(X\_{I\_t})=E(E(X\_{I\_t}\mid I\_t)) = E(\mu\_{I\_t}),$$
from which we have the desired equality. 


