Previous work has largely considered the fairness of image captioning systems through the underspecified lens of "bias. " In contrast, we present a set of techniques for measuring five types of representational harms, as well as the resulting measurements obtained for two of the most popular image captioning datasets using a state-of-the-art image captioning system. Our goal was not to audit this image captioning system, but rather to develop normatively grounded measurement techniques, in turn providing an opportunity to reflect on the many challenges involved. We propose multiple measurement techniques for each type of harm. We argue that by doing so, we are better able to capture the multi-faceted nature of each type of harm, in turn improving the (collective) validity of the resulting measurements. Throughout, we discuss the assumptions underlying our measurement approach and point out when they do not hold. CCS Concepts: • Social and professional topics → User characteristics; • Computing methodologies → Computer vision problems; Natural language processing.