Fourier transform inversion in the Alexiewicz norm

If $f\in L^1(\mathbb{R})$ it is proved that $\lim_{S\to\infty}\lVert f-f\ast D_S\rVert=0$,where$D_S(x)=\sin(Sx)/(\pi x)$istheDirichletkerneland$\lVert f\rVert = \sup_{\alpha<\beta}|\int_{\alpha}^{\beta}f(x),dx|$ is the Alexiewicz norm. This gives a symmetric inversion of the Fourier transform on the real line. An asymmetric inversion is also proved. The results also hold for a measure given by $dF$ where $F$ is a continuous function of bounded variation. Such measures need not be absolutely continuous with respect to Lebesgue measure. An example shows there is $f\in L^1(\mathbb{R})$ such that ${\lim }_{S\to \mathrm{\infty }}\parallel f-f\ast D_S{\parallel }_1\mathrm{\ne }0$.