Pettis measurability criterion for strong measurability

Strong measurability gives an essentially separable range. [given, L1, L4] Assume first that $f$ is strongly measurable, witnessed by $s_n$ and $N$ as in [L1]. The union of the finite ranges of the $s_n$ is countable. Its closed linear span $Y$ is separable by [L4], and every $f(\omega )$ with $\omega \mathrm{\notin }N$ is a norm limit of points of $Y$. Thus $f$ is essentially separably valued.

Strong measurability gives weak measurability. [given, L1, L2, L3] For $x^{\ast }\in X^{\ast }$, [L2] gives $x^{\ast }(s_n(\omega ))\to x^{\ast }(f(\omega ))$ off $N$. Each $x^{\ast }\circ s_n$ is scalar simple and measurable. A pointwise scalar limit is measurable off $N$, and [L3] makes its arbitrary values on subsets of $N$ measurable as well. Hence $f$ is weakly measurable.

Fix countable dense data for the reverse implication. [given, L4, choose] Conversely assume conditions 1 and 2. If $Y=\{0\}$, the constant zero simple functions converge to $f$ off $N$, so suppose $Y\mathrm{\ne }\{0\}$. By [L4] choose a sequence $(y_m)$ dense in $Y$ and a sequence $(z_k)$ dense in its unit sphere.

Construct a countable norming family. [A1, L2, L5, step 1.3] For each $k$, in the complex case define on the underlying real plane $\mathbb{C}{z}_k$ the norm-one real functional $u_k(a{z}_k)=\mathrm{Re}a$; in the real case use $u_k(a{z}_k)=a$ on $\mathbb{R}{z}_k$. Apply [L5] and [A1] to extend these simultaneously to real functionals $U_k$ on the underlying real space of $X$. In the complex case put $x_k^{\ast }(x)=U_k(x)-i{U}_k(ix)$; in the real case put $x_k^{\ast }=U_k$. Then $x_k^{\ast }\in X^{\ast }$, $\mathrm{\parallel }{x}_k^{\ast }\mathrm{\parallel }=1$, and $x_k^{\ast }(z_k)=1$. Consequently, for $y\in Y$,

Norm distances to fixed centres are measurable. [L3, step 2.1] For fixed $y\in Y$, step 2.1 and weak measurability give, off $N$, $\mathrm{\parallel }f-y\mathrm{\parallel }={\sup }_k\mathrm{\mid }{x}_k^{\ast }(f-y)\mathrm{\mid }$. The right side is the supremum of a countable family of measurable scalar functions. With any values assigned on $N$, [L3] therefore makes $\omega \mapsto \mathrm{\parallel }f(\omega )-y\mathrm{\parallel }$ measurable.

Build finite-valued nearest-centre approximants. [L1, step 1.3, step 3.1] For each $n\ge 1$ and $\omega \mathrm{\notin }N$, choose the least $m\in \{1,\dots ,n\}$ minimizing $\mathrm{\parallel }f(\omega )-y_m\mathrm{\parallel }$; put $s_n(\omega )=y_m$ there and $s_n=0$ on $N$. The finitely many tie-broken Voronoi cells are measurable by step 3.1, so $s_n$ is a measurable simple function. Density of $(y_m)$ gives $\mathrm{\parallel }{s}_n(\omega )-f(\omega )\mathrm{\parallel }\to 0$ for every $\omega \mathrm{\notin }N$.

Steps 1.1--1.2 prove the forward implication, and step 4.1 supplies the simple approximants required by [L1] for the reverse implication. The only non-finite choice is [A1]: it supplies the Hahn--Banach extensions in step 2.1 (and hence also covers their countable simultaneous selection).