In graph theory, a hypergraph is a generalization of a graph in which an edge can join any number of vertices.
A hypergraph is also called a set system.
in measure theory, a set system, is also defined as a set whose elements are themselves all sets, but the implication here is that the sets in question are augmented by the various operations, thus turning such sets of sets into algebraic structures.
This is a matter of conventions and expectations. For example, the set system of a hypergraph may have the property that it is a connected hypergraph. There are other properties that a set system may satisfy. For example, the set system of a topological space is required to satisfy certain properties. An alternative approach is to start with a base set and then to define an indicator function on the power set of the base set. The indicator function has the value $1$ if and only if the subset is an element of the corresponding set system. Using this approach we can include other functions on the power set. For example, the rank function of a matroid is required to satisfy certain properties. Similarly, in a meaurable space, the measure function is required to satisfy certain properties.