I don't understand the definition of $L^p(\partial\Lambda)$, for some bounded and open $\Lambda\subseteq\mathbb R^d$

Question

Let $d\in\mathbb N$, $\Lambda\subseteq\mathbb R^d$ be bounded and open and $p\ge 1$. In the definition of the trace operator, we consider the space $L^p(\partial\Lambda)$? But since $\partial\Lambda$ has Lebesgue measure zero and $L^p(\partial\Lambda)$ is a set of equivalence classes of Lebesgue almost equal functions, shouldn't it contain only the equivalence class of the zero function $\partial\Lambda\to\mathbb R$?