Agreement of the integral and L2 transforms

Choose the sequence $f_j$ of [F3]. Its terms are Schwartz, since every weighted derivative has compact support and is bounded. Thus [F1] identifies ${\mathcal{F}}_2{f}_j$ with the class of ${\widehat{f}}_j$. Also $\mathrm{\parallel }{\widehat{f}}_j-\widehat{f}{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }{f}_j-f{\mathrm{\parallel }}_1\to 0$ by [F2], whereas ${\mathcal{F}}_2{f}_j\to {\mathcal{F}}_{2}f$ in norm by [F1].

By [F4], a subsequence of the transform classes has measurable representatives tending a.e. to a representative $h$ of ${\mathcal{F}}_{2}f$. Those representatives and the continuous functions ${\widehat{f}}_j$ agree off a countable union of measurable null sets, so the corresponding subsequence of ${\widehat{f}}_j$ also tends to $h$ a.e. Step 1.1 gives its pointwise limit $\widehat{f}$ at every point by uniform convergence. Uniqueness of complex limits gives $h=\widehat{f}$ a.e. Countable choice is inherited from [F3], [F4] and Plancherel; no pointwise convergence of an arbitrary norm-convergent sequence is assumed.