What does this notation mean? $\mathbb{P}_{x\sim \mathcal{D}}$: Clarification in the definition of generalization error $R(h)$

Question

I was reading the book foundations of machine learning by Mehryar Mohri, Afshin Rostamizadeh and Ameet Talwalkar, and I came up with this definition of generalization error:

I dont understand what this means. I would like a more formal explanation of what they mean by the notation

$$ \mathbb{P}_{x\sim \mathcal{D}} [h(x)\neq c(x)]. $$

Maybe I'm wrong but does this mean that given a random variable $X$ taking values in the domain of $c$ with distribution $\mathcal{D}$ the generalization error is

$$ R(h) = \mathbb{P}[h(X)\neq c(X)] $$

Again, I just want a formal definition of what they are referring to. I understand the intuitive idea of the definition, they want to measure the probability of model $h$ being wrong when classifying.