I am currently in a revision of a paper. I have found something that I would only like to change if it is currently really notationally false because each change bears a few risks.

What I currently have is some notation for Bellman's equation, that is:

$$J(x)=\inf_{a} g(x,a) + \gamma \sum_{\tilde{x}\in\mathscr{X}} p_a(x,\tilde{x}) J(\tilde{x}),$$

where $x$ refers to a state variable, it might, for example, just be some scalar. $a$ is also some scalar. $\mathscr{X}$ is the set of possible states. A state could be a real number or integer, but for now I would like to keep this open. $0<\gamma<1$ is a discount factor, so that this infinite sum converges.

$p_a$ refers to the transition probability from some state $x$ to some state $\tilde{x}$. Let us assume for now that $p_a$ allows $\tilde{x}$ to take on an infinite amount of values (as opposed to some $p_a$ which is $0$ at most places but only $1$ for some specific $\tilde{x}$).

Despite these infinite transition states, $\tilde{x}$, is it still notationally acceptable to use the $\sum$ sign or would you consider it mandatory to use an integral notation?


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