Definition 5

— Let F be a Hausdorff locally convex space, $F'_s$ its dual equipped with the topology $\sigma(F', F)$. We denote by $\mathscr{L}_{F'_s}^\infty$ the vector space of mappings $f$ of T into $F'_s$, such that $f$ is scalarly $\mu$-measurable and is equal scalarly locally almost everywhere (for $\mu$) to a mapping of T into an equicontinuous subset of $F'$. We denote by $L_{F'_s}^\infty$ the quotient space of $\mathscr{L}_{F'_s}^\infty$ by the space of scalarly locally $\mu$-negligible mappings of T into $F'_s$.

When F satisfies the hypotheses of §1, No. 5, Prop. 13, the functions in $\mathscr{L}_{F'_s}^\infty$ are $\mu$-measurable for the weak topology $\sigma(F', F)$, but are not necessarily measurable for the strong topology on $F'$, even if F is a Banach space (§1, Exer. 17). Under the same conditions, the scalarly locally $\mu$-negligible mappings of T into $F'_s$ are identical to the locally $\mu$-negligible mappings of T into $F'_s$ (§1, No. 1, Remark 2).

When F is a separable normed space, the elements of $\mathscr{L}_{F'_s}^\infty$ are the mappings $f$ of T into $F'_s$ such that $f$ is scalarly $\mu$-measurable and $|f|$ is bounded in measure; one can then define a normed space structure on the space $L_{F'_s}^\infty$, by equipping it with the norm $N_\infty$ (Ch. IV, §6, No. 3).