Definition 16.1.1 ($B^{\infty}$ Spaces).label Let $(X, \cm, \mu)$ be a measure space and $E$ be a normed vector space, then the set $B^{\infty}(X; E)$ is the space of bounded $E$-valued strongly measurable functions on $X$, and the set $B^{\infty}(X)$ is the space of bounded complex-valued Borel measurable functions on $X$.