Abstract
The main purpose of this paper is to consider the strong law of large numbers for random sets in fuzzy metric space. Since many years ago, limited theorems have been expressed and proved for fuzzy random variables, but despite the uncertainty in fuzzy discussions, the nonfuzzy metric space has been used. Given that the fuzzy random variable is defined on the basis of random sets, in this paper, we generalize the strong law of large numbers for random sets in the fuzzy metric space. The embedded theorem for compact convex sets in the fuzzy normed space is the most important tool to prove this generalization. Also, as a result and by application, we use the strong law of large numbers for random sets in the fuzzy metric space for the bootstrap mean.
Highlights
E studies in this field began with the Puri and Ralescu [6] in 1983 for random sets in Banach space (Artstein et al [7] in 1975 and Cressie [8] in 1978 conducted studies on the SLLN in the Euclidean p-dimensional space)
When the uncertainty is due to fuzziness rather than randomness, as sometimes in the measurement of an ordinary length, it seems that the concept of a fuzzy metric space is more suitable. e concept of fuzzy metric space introduced by Kramosil and Michalek [19] and George and Veeramani [20] modified this concept
We prove an embedding theorem for random sets in fuzzy metric space
Summary
At first, we define the t-norm, the new generalized Hukuhara difference (T-difference), fuzzy metric space, and fuzzy normed space and give several lemmas and theorems in this space that will be used . E 3-tuple (X, M, ∗ ) is said to be a fuzzy metric space if M is a fuzzy set on X × X × (0, ∞) satisfying the following conditions for all x, y, z ∈ X and t, s > 0:. E 3-tuple (X, N, ∗ ) is said to be a fuzzy normed space if X is a vector space, ∗ is a continuous t-norm, and N is a fuzzy set on X × (0, ∞) satisfying the following conditions for every x, y ∈ X and t, s > 0:. A fuzzy metric MdH, which is induced by a fuzzy norm NdH, has the following properties for all A, B, C ∈ Kc(X) and every scalar λ ≠ 0:. By Lemma 1, Definition 9, and Lemma 4 [29], it is easy to show that the result is established; for example, for (2), we have
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