The connection between ordinary and soft $\sigma$-algebras with applications to information structures

The paper presents a novel analysis of interrelations between ordinary (crisp) $\sigma$-algebras and soft $\sigma$-algebras. It is known that each soft $\sigma$-algebra produces a system of crisp (parameterized) $\sigma$-algebras. The other way round is also possible. That is to say, one can generate a soft $\sigma$-algebra from a system of crisp $\sigma$-algebras. Different methods of producing soft $\sigma$-algebras are discussed by implementing two formulas. It is demonstrated how these formulas can be used in practice with the aid of some examples. Furthermore, we study the fundamental properties of soft $\sigma$-algebras. Lastly, we show that elements of a soft $\sigma$-algebra contain information about a specific event.