In this talk, I will talk about a useful tool in extremal value theory. Extremes of a stationary sequence can be fitted into the framework of random sup-measures, which was pointed out by O'Brien et al in the 1980s. I will explain the random sup-measures, and see their applications in the heavy-tailed settings. A specific random sup-measure arising from a heavy-tailed stationary process with long-range dependence will be explored. I will also explain how the dependence structures are reflected in the limiting theorem.