  We introduce a comprehensive framework for analyzing convergence rates for
infinite dimensional linear programming problems (LPs) within the context of
the moment-sum-of-squares hierarchy. Our primary focus is on extending the
existing convergence rate analysis, initially developed for static polynomial
optimization, to the more general and challenging domain of the generalized
moment problem. We establish an easy-to-follow procedure for obtaining
convergence rates. Our methodology is based on, firstly, a state-of-the-art
degree bound for Putinar's Positivstellensatz, secondly, quantitative
polynomial approximation bounds, and, thirdly, a geometric Slater condition on
the infinite dimensional LP. We address a broad problem formulation that
encompasses various applications, such as optimal control, volume computation,
and exit location of stochastic processes. We illustrate the procedure at these
three problems and, using a recent improvement on effective versions of
Putinar's Positivstellensatz, we improve existing convergence rates.
