Abstract
We introduce the idea of level sets which are crisp sets associated with a fuzzy set. We show how we can represent a fuzzy set using a collection of level sets; this allows us to extend set functions, such as probability and other set measures, to fuzzy subsets. We emphasize that this extension is itself a fuzzy subset. We next introduce Pythagorean fuzzy subsets and discuss the OWA and Choquet aggregation of these sets. We provide a formulation for level sets associated with a Pythagorean fuzzy subset that we use to provide a representation of a Pythagorean fuzzy set in terms of level sets. With the aid of Zadeh's extension principle, we use this level set representation to provide for an extension of set measures to Pythagorean fuzzy sets. We then suggest an alternative approach to extending set measures to Pythagorean fuzzy sets using the Choquet integral.