The authors introduce several convergence concepts of fuzzy sets such as convergent in measure, convergent uniformly, \(\mu\)-uniform convergence and \(L^ p\)-convergent. After defining these concepts, the authors construct an embedding of the space of fuzzy sets in a normed linear space. Then based on this embedding of the space of fuzzy sets in a normed linear space, the authors define the \(L^ p\)-convergence concept. With the help of these, the authors have proved the very interesting property which states that the space of fuzzy sets under the embedding is a complete metric space.