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I don't think this is silly. For example, if $G=\mathbb R$ then for each $t\in\mathbb R$ consider $$ f_{t,n} = \frac{n}{2} \chi_{[t-1/n,t+1/n]} \in L^1(\mathbb R). $$ Then each $f_{t,n}$ is a unit vector in $L^1(\mathbb R)$. Then given $F\in L^\infty(G)$, we define $$\tilde F(t) = \lim_n \langle F,f_{t,n} \rangle = \lim_n \frac{n}{2} \int_{t-1/n}^{t+1/n} F(s) \ ds$$ Maybe this limit doesn't actually exist-- but the sequence is bounded, so just force it to converge via an ultra-filter limit, or similar. I think you have just described an abstract version of this construction. The point is that $\tilde F$ is little more than a function $\mathbb R\rightarrow\mathbb C$ which is bounded; I don't see why it need have any continuity or measurability properties...