Let $(\Omega, \mathcal{A}, \mu)$ be an arbitrary measure space, and $E$ a convex subset of a Hausdorff topological vector space. Any function from $\Omega\times E$ into $( - \infty, +\infty]$ is be called integrand. An integrand $g : \Omega \times E \to ( - \infty, +\infty]$ is said to be respectively convex, affine, (sequentially) lower semicontinuous, (sequentially) continuous, or (sequentially) inf-compact if for every $\omega\in \Omega$ the function $g(\omega, .)$ on $E$ has the corresponding property (recall that $g(\omega, .)$ is (sequentially) inf-compact on $E$ if for every $\beta \in\mathbb{R}$ the set of all $x \in E$ such that $g(\omega, x) \leq \beta$, is (sequentially) compact). Thus, these adjectives refer only to the behavior of the integrand in its second variable.
Let $h: \Omega \times E \to [0, + \infty]$ be given nonnegative convex sequentially inf-compact integrand and let $\mathcal{F}$ be a set of affine sequentially continuous integrands $a: \Omega \times E \to\mathbb{R}$. A function $f: \Omega \to E$ is defined to be $\mathcal{F}$-scalarly measurable if: $$ \omega\to a\big( \omega,f(\omega) \big)\text{ is measurable on }\Omega \text{ for every }a \in \mathcal{F} $$ I don't understand this last definition definition. Does it have a relation with a scalarly measurable function?