If $\mathfrak{X}$ is a Banach space, a function $T: \mathbb{R} \to \mathcal{L}(\mathfrak{X})$ is defined to be
- uniformly measurable if it is an a.e. norm limit of a sequence of countably valued functions from $\mathbb{R}$ to $\mathcal{L}(\mathfrak{X})$
- strongly measurable if, for each $x \in \mathfrak{X}$, $t \mapsto T(t)x$ is an a.e. norm limit of a sequence of countably valued functions from $\mathbb{R}$ to $\mathfrak{X}$
- weakly measurable if, for each $x \in \mathfrak{X}$ and $\ell \in \mathfrak{X}^*$, $t \mapsto \ell(T(t) x)$ is a measurable function from $\mathbb{R}$ to $\mathbb{C}$
If I had to guess from the terminology, I would have said that uniform, strong, and weak measurability are just measurability with respect to the Borel $\sigma$-algebras generated by the uniform, strong, and weak topologies on $\mathcal{L}(\mathfrak{X})$. Would that be equivalent? Or are all those Borel $\sigma$-algebras the same for some reason that I'm not seeing? Or is the resulting integration theory not as satisfactory? The books I'm reading (Hille/Phillips and Dunford/Schwartz) don't have any discussion of why we don't define measurability in (what seems to me) the natural way.