Definition 25.1.2 (Space of Measurable Functions).label Let $(X, \cm)$ and $(Y, \cn)$ be measurable spaces, then the set $\mathscr{L}^{0}(X, \cm; Y) = \mathcal{L}^{0}(X; Y)$ is the space of measurable functions from $X$ to $Y$.

For any measure $\mu$ on $(X, \cm)$, the space $L^{0}(X, \cm, \mu; Y) = L^{0}(X; Y)$ is the space of measurable functions from $X$ to $Y$, modulo almost everywhere equality.