Need help proving policy improvement theorem for epsilon greedy policies

Question

I'm slightly confused about the proof epsilon greedy policy improvement. This is a part of the proof:

$\begin{aligned} q_{\pi}\left(s, \pi^{\prime}(s)\right) &=\sum_{a} \pi^{\prime}(a \mid s) q_{\pi}(s, a) \\ &=\frac{\varepsilon}{|\mathcal{A}(s)|} \sum_{a} q_{\pi}(s, a)+(1-\varepsilon) \max _{a} q_{\pi}(s, a) \\ & \geq \frac{\varepsilon}{|\mathcal{A}(s)|} \sum_{a} q_{\pi}(s, a)+(1-\varepsilon) \sum_{a} \frac{\pi(a \mid s)-\frac{\varepsilon}{|\mathcal{A}(s)|}}{1-\varepsilon} q_{\pi}(s, a) \end{aligned}$

I'm stuck on the very first line itself. Why is $q_{\pi}\left(s, \pi^{\prime}(s)\right) =\sum_{a} \pi^{\prime}(a \mid s) q_{\pi}(s, a)$? I've looked at other similar questions on this site, but they dont seem to address this line of the proof.

Answer 1

Sutton describes it as a natural definition.

If we are at state $s$, we take action $a$ with probability $\pi'(a|s)$ according to policy $\pi'$ and we compute the expected value.