TY - CHAP

T1 - The space L1∩L∞

AU - Rubshtein, Ben Zion A.

AU - Grabarnik, Genady Ya

AU - Muratov, Mustafa A.

AU - Pashkova, Yulia S.

N1 - Publisher Copyright:
© Springer International Publishing Switzerland 2016.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - The space L1 ∩ L∞, which we study in this chapter, consists of all bounded integrable functions equipped with the norm ∥·∥L1 ∩ L∞= max(∥·∥L1, ∥·∥L∞). We show that. (L1 ∩ L∞; ∥·∥L1 ∩ L∞) is a symmetric space and describe the closure L0∞ of L1 ∩ L∞ in L∞. Given two equimeasurable functions f and g, we treat an approximation of g in the L1 ∩ L∞-norm by shifted functions f o θ, where θ is a measure-preserving transformation. Step functions and integrable simple functions are applied for this purpose.

AB - The space L1 ∩ L∞, which we study in this chapter, consists of all bounded integrable functions equipped with the norm ∥·∥L1 ∩ L∞= max(∥·∥L1, ∥·∥L∞). We show that. (L1 ∩ L∞; ∥·∥L1 ∩ L∞) is a symmetric space and describe the closure L0∞ of L1 ∩ L∞ in L∞. Given two equimeasurable functions f and g, we treat an approximation of g in the L1 ∩ L∞-norm by shifted functions f o θ, where θ is a measure-preserving transformation. Step functions and integrable simple functions are applied for this purpose.

UR - http://www.scopus.com/inward/record.url?scp=85006312763&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-42758-4_3

DO - 10.1007/978-3-319-42758-4_3

M3 - Chapter

AN - SCOPUS:85006312763

T3 - Developments in Mathematics

SP - 29

EP - 40

BT - Developments in Mathematics

PB - Springer New York LLC

ER -