# Is it incorrect to have a sum of an infinite weighted set?

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?