N-functions, Orlicz functions and Orlicz classes and Orlicz spaces generated by N-functions and Orlicz functions have been studied by many mathematicians as in [24],[17],[28],[18],[25],[3],[4],[5],[26],[11],[20],[9]. Musielak-Orlicz functions and Musielak-Orlicz spaces generated by Musielak-Orlicz functions have been originated and developed by [23],[22],[21] where f ? LMO (?,?,?) if and only if ??? MO(t,f(t))d? < ?. Their properties have been studied by [13],[16],[8],[14],[15],[16],[33] and their applications can be found in differential equations [7],[10], fluid dynamics [29],[31], statistical physics[1],integral equations [17], image processing [2],[6],[12] and many other applications [27]. So, because such increasingly importance to these concepts in the modeling of modern materials, we want to investigate properties, calculus and basic approx- imations of Musielak N-functions and Musielak-Orlicz functions and their Musielak-Orlicz spaces using the measure theory where this will help us to consider ??almost everywhere property, supremum, infimum, limit, convergence and basic convergence of Musielak N-functions, Musielak-Orlicz functions and Musielak-Orlicz spaces generated by them by functioning facts and results of the measure theory and getting advantages from that to consider these concepts. The concept of Musielak N -function M (t, u) is a generalization to the concept of N-function M(u), where M(t,u) may vary with location in space, whereas the Musielak-Orlicz function MO(t,u) is a generalization to the concept of Orlicz functions O(u), where MO(t,u) may vary with location in space. The Musielak-Orlicz function M O(t, u) is defined on ? × [0, ?) into [0, ?) where for ??a.e. t ? ?,MO(t,.) is Orlicz function of u on [0,?) and for each u ? [0,?),MO(.,u) is a ??measurable function of t on ? and (?, ?, ?) is a measure space [3],[15]. So, we are going to define the Musielak N -function M(t,u) on ?×R into R in similar way, that is, for ??a.e. t ? ?,M(t,.) is N-function of u on R and for each u ? R, M (., u) is a ??measurable function of t on ? and (?, ?, ?) is a measure space. The novelty to define the Musielak N -function M (t, u) by this way is to get benefits from the results of the measure theory and use them to consider properties, calculus, and basic convergence of Musielak N-functions and Musielak-Orlicz spaces generated by them and the relationship between Musielak N-functions and Musielak-Orlicz functions and their Musielak-Orlicz spaces generated by them where this will give us more flexibility to pick a suitable measurable set ? and then the functional ??? M(t,?f(t)?BS)d? defined on it as we will see in section 3. So, the paper is organized as follows. Definition of Musielak N-function, developing preliminaries results about the equivalent definition of Musielak N-function and studying continuity of Musielak N-function are
In this paper, the concept of Musielak N-functions and Musielak-Orlicz spaces generated by them well be introduced. Facts and results of the measure theory will be applied to consider properties, calculus and basic approximation of Musielak N-functions and their Musielak-Orlicz spaces. Finally, the relationship between Musielak N-functions and Musielak-Orlicz functions and thier Musielak-Orlicz spaces will be considered using facts and results of the measure theory too.
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arXiv:1806.07310v1 [math.FA] 19 Jun 2018
introduced in section 2. Definition of Musielak-Orlicz space generated by a Musielak N-function, and using facts and results of the measure theory to study properties, calculus and basic approximation of Musielak N-functions and Musielak-Orlicz spaces generated by them in section 3. The relationship between Musielak N-functions and Musielak-Orlicz functions and Musielak-Orlicz spaces generated by them respectively using facts and results of the measure theory are introduced also in section 4. Examples of Musielak N-functions and Musielak-Orlicz functions that are not Musielak N-functions will be in section 5. The conclusion will be in section 6.