The Haar Integral, Edition: 1st American Edition by Leopoldo Nachbin

By Leopoldo Nachbin

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3. If g. / D 1= for 0 < < 0 , and 0 , then w is nonstrict and p g. x; y/g. 4. For g. x; y/g. 5. If g. / D 1 for 0 < < 0 , and g. x; y/, where • is the discrete metric on X. 6. x; y/ with g0 D inf f > 0 W g. / Ä g. 3. 1. If is a classical modular on a real linear space X (cf. Sect. ˛x/ D 0g is called the modular space (with zero as its center). The modular space X is a linear subspace of X, and the functional j j W X ! 4) jcn xn cxj ! 0 as n ! 1 whenever cn ! c in R and jxn xj ! 0 as n ! 1 (where xn 2 X for n 2 N).

G 0 /0 Ä g0 . 8. x; y/ for all x; y 2 X. In particular, if w and w are (pseudo)modulars on X such that wC0 D wC0 or 0 0 X. 0 D w 0 , then dw D dw on X We conclude that the right and left regularizations of a (pseudo)modular w on X provide no new modular spaces as compared to Xw , Xw0 and Xwfin (cf. Sect. 1) and no new (pseudo)metrics as compared to dw0 . Yet, in Sect. 5, we establish the existence of continuum many (equivalent) metrics on the modular space Xw . x; y/ at the level of the map g 7!

Extended metrics, also called generalized metrics, were studied by Jung [50] and Luxemburg [67] in connection with an extension of Banach’s Fixed Point Theorem from [4]. The interpretation of a modular as a generalized velocity field was initiated by Chistyakov in [26, 28]. 2. 1 of (metric) modular w on a set X appeared implicitly in Chistyakov [18, 19] in connection with the studies of (bounded variation and the like) selections of set-valued mappings, and multivalued superposition operators. Explicitly and axiomatically, (pseudo)modulars were introduced in Chistyakov [22], and their main properties were established by the author in [23–25].

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