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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!

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    $\begingroup$ 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. $\endgroup$ – Nate Eldredge Oct 10 '14 at 18:16
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    $\begingroup$ Is it even true for $G = B = \mathbb{R}$? $\endgroup$ – Nate Eldredge Oct 10 '14 at 18:19
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    $\begingroup$ @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? $\endgroup$ – Transcendental Oct 10 '14 at 19:43
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    $\begingroup$ Using the Borel isomorphism theorem, the problem should be reducible to $B=\mathbb{R}$. $\endgroup$ – Michael Greinecker Oct 29 '14 at 23:25

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