# Where does the expected value in the restatement of the pseudoregret come from?

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.)

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.