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 theeverywherepointwise 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!