Probabilistic modular spaces and linear operators (Q1038712)

Let \(X\) be a real vector space and let \(\Delta\) be the set of all non-decreasing functions \(f:\mathbb{R}\to \mathbb{R}^+_0\) with \(\text{inf\,}f(x)= 0\), \(\sup f(x)= 1\). Probabilistic modular spaces \((X,\mu)\) are considered, where \(\mu: X\to \Delta\) satisfies suitable conditions. In the case \(\mu(0)= 0\), \(\mu(x)= 1\) for \(x\neq 0\), \((X,\mu)\) is a usual modular space. The paper contains considerations concerning continuity and boundedness of linear operators between probabilistic modular spaces.