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.