Specification formalisms like LTL specify properties for infinite traces whereas monitors only observe finite prefixes. These can be conclusive, however, in general they are not. Restricting monitors to only two possible conclusions is therefore inappropriate in many cases and known approaches distinguish only between a small number of verdicts. We extend the setting to an infinite, continuous domain that enables us to further discriminate runs and allows for more detailed appraisals in presence of inconclusive information. The concept of continuous space induces new notions of monitorability that we study and relate to well known classes.