Generalized symmetric Orlicz $F$-spaces

The paper is devoted to the consideration of a class of examples of symmetric $F$-spaces of measurable functions on spaces with finite or infinite $\sigma$-finite non-atomic measure. We do not assume separability conditions for the measure. Moreover, we use the correspondence between symmetric spaces on general spaces with a measure and their "standard" copies on a semiaxis or segment. Every symmetric $F$-space is a linear metric space. The definition of linear metric spaces was first given by Frechet in 1926. Later, Stefan Banach and his students proved the basic facts of the theory of linear metric and Banach spaces. At the beginning, normalized and locally convex spaces were studied. The development of the theory of integral operators and the theory of random processes aroused interest in the theory of non-locally convex spaces. The theory of non-locally convex spaces has been intensively developed. New applications have been obtained in probability theory, integral operator theory, and analytic function theory. Recently, many papers have appeared on quasi-normalized spaces, non-interpolation spaces, and spaces that do not have the property of local convexity. A general view of such spaces led to the study of $F$-spaces of measurable functions on spaces with finite and infinite measure. In the works of E. M. Semenov, connected with the theory of interpolation of linear operators in spaces of measurable functions, symmetric Banach spaces were investigated, which in the foreign literature were called rearrangement invariant spaces. The theory of symmetric spaces has been intensively developing over the last century, contains many interesting and profound results and has important applications in various fields of function theory and functional analysis, in particular in ergodic theory, harmonic analysis and mathematical physics. Therefore, it is natural to study symmetric $F$-spaces of measurable functions. In this paper, as well as for symmetric Banach spaces, for symmetric $F$-spaces, the concept of equimeasurablity is introduced. It is proved that each class of equimeasurable symmetric $F$-spaces contains a standard symmetric space, while all equimeasurable standard symmetric spaces coincide. Classical examples of symmetric Banach spaces of measurable functions are the Banach Orlicz spaces. The Orlicz spaces are described in detail in the paper M. A. Krasnoselskii and Ya. B. Rutitzkii "Convex functions and Orlicz spaces" (1961). When constructing the Orlicz space, the so-called $N$-function, which is convex, plays an essential role. The paper considers a class of examples of $F$-spaces called generalized Orlicz spaces, which are constructed by functions that are not generally convex, but have only the monotonicity property.