Tensor products are dense in Bochner L2 under scalar and spatial density

Fix $U\in L_{\rho }^2(V)$ and $\epsilon >0$. By [F1] and [F5], choose a simple approximation sequence and replace it and $U$ by zero on their measurable null exceptional set. Write $S_n$ for the resulting simple maps, which converge everywhere to the Borel representative $U$. Set $T_n=S_n{\mathbf{1}}_{\{\mathrm{\parallel }{S}_n{\mathrm{\parallel }}_V\le 2\mathrm{\parallel }U{\mathrm{\parallel }}_V\}}$. The truncation sets are measurable and $T_n$ is finite-valued simple. At $U=0$ it is zero; at every other point it eventually equals $S_n$, so $T_n\to U$. Since $\mathrm{\parallel }{T}_n-U{\mathrm{\parallel }}_V^2\le 9\mathrm{\parallel }U{\mathrm{\parallel }}_V^2$, [F2] gives $\mathrm{\parallel }{T}_n-U{\mathrm{\parallel }}_{L_{\rho }^2(V)}\to 0$. Choose one such map $S={\sum }_{k=1}^m{\mathbf{1}}_{E_k}{v}_k$ with error below $\epsilon \mathrm{/}2$, taking $m\ge 1$ by allowing a zero value. It remains to approximate its finitely many terms with total error below $\epsilon \mathrm{/}2$.

For each $k$, use [F3] and [F4] to choose $s_k\in \mathcal{S}$ and $w_k\in V_0$ with $\mathrm{\parallel }{\mathbf{1}}_{E_k}-s_k{\mathrm{\parallel }}_{L^2(\rho )}\mathrm{\parallel }{v}_k{\mathrm{\parallel }}_V+\mathrm{\parallel }{s}_k{\mathrm{\parallel }}_{L^2(\rho )}\mathrm{\parallel }{v}_k-w_k{\mathrm{\parallel }}_V<\epsilon \mathrm{/}(2m)$. Such choices are possible by first making the scalar error arbitrarily small, which bounds $\mathrm{\parallel }{s}_k{\mathrm{\parallel }}_2\le \mathrm{\parallel }{\mathbf{1}}_{E_k}{\mathrm{\parallel }}_2+\mathrm{\parallel }{\mathbf{1}}_{E_k}-s_k{\mathrm{\parallel }}_2$, and then making the vector error small. The product norm estimate $\mathrm{\parallel }{\mathbf{1}}_{E_k}{v}_k-s_k{w}_k{\mathrm{\parallel }}_{L_{\rho }^2(V)}\le \mathrm{\parallel }{\mathbf{1}}_{E_k}-s_k{\mathrm{\parallel }}_2\mathrm{\parallel }{v}_k{\mathrm{\parallel }}_V+\mathrm{\parallel }{s}_k{\mathrm{\parallel }}_2\mathrm{\parallel }{v}_k-w_k{\mathrm{\parallel }}_V$ follows pointwise from the triangle inequality and then from scalar square-integral estimation. Each product is strongly measurable by [F5], so these are legitimate Bochner-space norms.

The finite sum ${\sum }_{k=1}^m{s}_k{w}_k$ belongs to the displayed algebraic tensor span, and the triangle inequality with step 2.1 gives distance less than $\epsilon \mathrm{/}2$ from $S$. Together with step 1.1 its distance from $U$ is less than $\epsilon$. Since $U$ and $\epsilon$ were arbitrary, the span is dense.