Define: $\Delta A(t) = A(t+1)-A(t)$ and let $$ \Delta C = \sum^{T-1}_{t=0} [~~H(\pi(t+1)~|~\mu(t+1))~~ - ~~H(\pi(t)~|~\mu(t+1))~~] $$ Where $H (\pi(t+1)|\mu(t+1)) = \sum^n_{i=1}\pi_i(t) \log \frac{\pi_i(t)}{\mu_i(t+1)}$ is the relative entropy.
I am confused about the following:
What is the definition of $C$
What is $C$ a function of?
attempt I believe this implies that: $$ C(s) = \sum_{t=1}^{T-1}H(s~|~\mu(t+1)) $$
Having taken a look at the paper you mentioned in the comments I would rewrite the beginning of your post as follows:
Define: $\Delta A(t) = A(t+1)-A(t)$ and let $$S = \sum^{T-1}_{t=0} \Delta C(t)$$ where $$\Delta C(t) = ~~H(\pi(t+1)~|~\mu(t+1))~~ - ~~H(\pi(t)~|~\mu(t+1))~~$$ and $H (\pi(t+1)|\mu(t+1)) = \sum^n_{i=1}\pi_i(t) \log \frac{\pi_i(t)}{\mu_i(t+1}$ is the relative entropy.
Here we see that $\Delta C(t)$ is a function of $t$ and the sum $S$ (what you originally called $\Delta C$) is a number. Alternatively, if you would like to think of the $T$ as varying, then you could consider $S$ to be a function of $T$: $$S(T) = \sum^{T-1}_{t=0} \Delta C(t)$$