Chapter 1 - The superposition operator in the space S

Summary

In this chapter we study the superposition operator Fx(s) = f(s,x(s)) in the complete metric space S of measurable functions over some measure space Ω. First, we consider some classes of functions f which generate a superposition operator F from S into S; a classical example is the class of Carathéodory functions, a more general class that of Shragin functions.

As a matter of fact, there exist functions f, called “monsters”, which generate the zero operator Fx ≡ θ, but are not measurable on Ω × ℝ, and hence are not Carathéodory functions; this disproves the old-standing Nemytskij conjecture. On the other hand, we show that a function which generates a continuous superposition operator (in measure) is “almost” a Carathéodory function.

We give a necessary and sufficient condition for the function f to generate a bounded superposition operator F in the space S. In particular, this conditions holds always if f is a Carathéodory function. On the other hand, we show that the superposition operator F is “never” compact in the space S, except for the trivial case when F is constant.

Finally, we consider superposition operators which are generated by functions f with special properties (e.g. monotonicity), and characterize the points of discontinuity of such operators.

The space S

Let Ω be an arbitrary set, M some σ-algebra of subsets of Ω (which will be called measurable in what follows), and µ a countably additive and σ-finite measure on M.