Extension of measures on two disjoint rings

On measure relatively to fuzzy σ–algebra

“…Many authors was studied σ-algebra, see [5], [6] and [7]. The notion of measure is studied in [8] and [9] as a set function 𝜓: ℋ ⟶ [0 , ∞] such that 𝜓 (⋃ E 𝑘 𝑘 ) = ∑ 𝜓 (E 𝑘 ) 𝑘 , whenever E 1 , E 2 , … form a finite or countably infinite collection of disjoint sets in ℋ . Zadeh [10] in 1965 first introduced the concept of the fuzzy set where 𝒳 is non-empty set, then a fuzzy set F in 𝒳 is defined as a set of ordered pairs {(ω, 𝓋 F (𝜔)) : 𝜔 ∈ 𝒳 } where 𝓋 F : 𝒳 → [0 , 1] is function such that for every 𝜔 ∈ 𝒳, 𝓋 F (𝜔) represent the degree of membership of 𝜔 in F. Ahmed et al [11] first introduced the concept of fuzzy σ −algebra and fuzzy algebra, where 𝒳 be a non-empty set.…”