Stopping time, why $E[S_{\nu}]=\mu E[\nu]$?

Question

Let $Y_1,Y_2,....$ be iid random variables with mean $\mu$, let $S_n =\sum_{i=1}^{n}Y_i$. Let $F_n=\sigma(Y_1,...,Y_n)$. Let $\nu$ be a stopping time s.t. $E[\nu]<\infty$. Why $E[S_{\nu}]=\mu E[\nu]$?

Answer 1

Using the identity $E[Y]=E[E[Y|X]]$:

$$E[S_{\nu}]=E\left[\sum_{i=1}^{\nu}Y_i\right] = E\left[E\left[\sum_{i=1}^{\nu}Y_i\Bigg|\nu\right]\right] = E\left[\nu E[Y_k|\nu]\right]= E[\nu \mu]=\mu E[\nu]$$

As

$$E\left[\sum_{i=1}^{\nu}Y_i\Bigg|\nu\right] = \nu E[Y_k|\nu]$$

because once the value of $\nu$ is known, the stochastic sum is just an ordinary sum. And since all $Y_k$ are identically distributed and their distribution is independent of $\nu$, we have that for all $k$

$$E[Y_k|\nu]=E[Y_1|\nu]=E[Y_1] = \mu$$