National forest inventories are based on probabilistic sampling designs. It is common practice to randomly select a sample of points in a continuum (the territory under study) and then to define fixed-shape supports (e.g., plots or polygons) from these points to perform the survey on the field on the population of trees; see for example Vidal et al. (2016) for a worldwide overview of sampling designs used in forest inventories. Although the sampling design may be formalized in several manners (e.g., Eriksson, 1995), the infinite population approach (Stevens and Urqhart, 2000; Barabesi, 2003; Mandallaz, 2007; Gregoire and Valentine, 2007) is arguably the simplest device for inference. Inference may be performed directly from the sampled population, which is straightforward by using the theory of continuous Horvitz-Thompson (HT) estimation (Cordy, 1993), both in terms of point estimation and variance estimation (Chauvet et al., 2023). In order to produce reliable estimators, some important properties are needed for continuous sampling designs. The HT-estimator should be consistent and asymptotically normally distributed, with consistent variance estimators for interval estimation. These properties have not been much considered in the literature, with the exception of Barebesi and Franceschi (2011) and Barabesi et al. (2012). In this work, we derive these properties for general continuous sampling designs, under mild conditions.