We characterize some large cardinal properties, such as μ-measurability and P2(κ)-measurability, in terms of ultrafilters, and then explore the Rudin-Keisler (RK) relations between these ultrafilters and supercompact measures on Pκ(2κ). This leads to the characterization of 2κ-supercompactness in terms of a measure on measure sequences, and also to the study of a certain natural subset, Fullκ, of Pκ(2κ) whose elements code measures on cardinals less than κ. The hypothesis that Fullκ is stationary (a weaker assumption than 2κ-supercompactness) is equivalent to a higher order Löwenheim-Skolem property, and settles a question about directed versus chain-type unions on Pκλ.
