Definition of the periodic $L^p$ space on torus

In his Real Analysis, Folland uses the notation $L^p({\mathbb T}^n)$ (where $\mathbb{T}^n$ denotes the n-dimensional torus) is used before Hausdorff measure is introduced. (See for instance Chapter 8: Elements of Fourier Analysis) It is unclear to me that how this space is defined without referring to Hausdorff measures.

What is the "usual" definitions for $L^p({\mathbb T}^n)$? Would anyone also come up with some references?