Approximating a measurable function from a second-countable, locally compact Hausdorff group to a separable Banach space

Let $G$ be a second-countable, locally compact Hausdorff group and $B$ a separable Banach space.

We say that a function $f: G \to B$ is Bochner-measurable if and only if it is the everywhere pointwise limit of a sequence of simple measurable functions.

We say that a function $f: G \to B$ is Borel-measurable if and only if the pre-image of every open subset of $B$ is a Borel subset of $G$.

According to Propositions E.1 and E.2 of Donald Cohn’s book Measure Theory, Bochner-measurability and Borel-measurability coincide because $B$ is assumed to be separable. As these two notions of measurability coincide, we will simply label a function that is measurable in either sense as ‘measurable’.

Question: Is it true that a measurable function $f: G \times G \to B$ is the everywhere pointwise limit of a sequence $(\sigma_{n})_{n \in \mathbb{N}}$ of simple measurable functions such that each $\sigma_{n}$ has the form $$\sum_{k = 1}^{m} \chi_{E_{k} \times F_{k}} \cdot b_{k},$$ where $b_{k} \in B$, and $E_{k}$ and $F_{k}$ are Borel subsets of $G$ for each $k \in \{ 1,\ldots,m \}$?

Thank you all for your help!

• Interesting question. It is sufficient to consider the case $f = 1_A$ where $A \subset G \times G$ is Borel, since linear combinations of those give you all simple functions on $G \times G$, and every measurable function is a uniform limit of simple functions. I think it is true, by a monotone class argument, that the measurable functions are the smallest set of functions containing all $\sum b_k \chi_{E_k \times F_k}$ and closed under pointwise convergence of sequences. But your condition is more like a Baire class question. – Nate Eldredge Oct 10 '14 at 18:16
• Is it even true for $G = B = \mathbb{R}$? – Nate Eldredge Oct 10 '14 at 18:19
• @NateEldredge: Hi Nate. I’ve investigated $G = B = \mathbb{R}$, and so far, I haven’t encountered any counterexamples. I’ve also been thinking about using a monotone-class argument, but I’m not sure how one builds a Borel subset of $G \times G$ by a transfinite application of the axioms of a monotone class. Do we have an analog of the Borel hierarchy for a monotone class that tells us how members of the class are constructed? – Transcendental Oct 10 '14 at 19:43
• Using the Borel isomorphism theorem, the problem should be reducible to $B=\mathbb{R}$. – Michael Greinecker Oct 29 '14 at 23:25