In the theory of Bochner integration (taking definitions from Yosida), letting $(E, \mathcal{F},\mu)$ a complete measure space and $X$ a Banach space, there's a simple result (attributed to Bochner) saying that $f: E \to X$ is integrable (meaning there exist simple functions non zero on sets of finite measure $\phi_j$ s.t. $\int \|\phi_j - f\| \to 0$) iff $f$ is strongly measurable (pointwise a.e. limit of simple functions) and $\int \|f\| < \infty$. There's also a (variation on a) result due Pettis saying $f$ is strongly measurable iff $f$ is measurable with respect to the Borel $\sigma$-algebra on $X$ and $f$ is almost separably valued.
These results naturally raise the question of the sharpness of the above result, namely if the requirement of $f$ being almost separably valued is necessary or if it comes out for free. Made precise, the question is:
- Does there exist $f: E \to X$ measurable with respect to the Borel $\sigma$-algebra on $X$ s.t. $\int \|f\| < \infty$ but s.t. $f$ is not almost separably valued?
This is equivalent to asking whether if $f$ is measurable and the norm of $f$ has finite integral, is $f$ then strongly measurable?