ABSTRACT
The modular spaces ℓ p n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781498762618/4882778a-ba8d-4a3b-a129-38b97b5fc627/content/eq3284.tif"/> introduced in Section 1.4 and the space L p ( Ω ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781498762618/4882778a-ba8d-4a3b-a129-38b97b5fc627/content/eq3285.tif"/> defined in Section 1.5 are but particular instances of the same mathematical concept. The key observation to study their commonalities rather than emphasizing their differences is that the modulars ρ p n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781498762618/4882778a-ba8d-4a3b-a129-38b97b5fc627/content/eq3286.tif"/> and ρ p https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781498762618/4882778a-ba8d-4a3b-a129-38b97b5fc627/content/eq3287.tif"/> introduced in Section 1.4 and in Section 1.5, respectively entail a higher degree of complexity than that inherent to the mere definition of modular given in Section 1.3. Specifically, from now on we will focus on a detailed exploration of the intrinsic measure-theoretic character of both ρ p n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781498762618/4882778a-ba8d-4a3b-a129-38b97b5fc627/content/eq3288.tif"/> and ρ p https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781498762618/4882778a-ba8d-4a3b-a129-38b97b5fc627/content/eq3289.tif"/> . To the effect of precisely describing the specialness of the particular class of modulars we have in mind, some terminology must be introduced.
