I am working through the chapter on Fourier Transforms in Rudin's Real and Complex Analysis, and I am having difficulty understanding a property of the Fourier Transform. Rudin defines the Fourier Transform as follows:
Here, the measure m is equal to the Lebesgue measure divided by $\sqrt{2\pi}$. He then gives the following theorem:
He states that properties (a), (b), (d), and (e) all follow by direct substitution into the definition. I did this for (a), (b), and (e) and was able to verify those 3, but I am having difficulty with (d). My attempt is as follows:
If $g(x) = \overline{f(-x)}$, then $$\hat{g}(t) = \int_{-\infty}^{\infty}g(x) e^{-ixt}dm(x) = \int_{-\infty}^{\infty}\overline{f(-x)}e^{-ixt}dm(x) = \int_{-\infty}^{\infty}\overline{f(-x)e^{ixt}}dm(x) = \overline{\int_{-\infty}^{\infty}f(-x)e^{ixt}dm(x)} = \overline{-\int_{-\infty}^{\infty}f(x)e^{-ixt}dm(x)} = \overline{-\hat{f}(t)}$$
The problem with my attempt is I can't get rid of the negative sign that comes in the last step when I change variables from $-x$ to $x$. Can somebody elaborate on what I am missing?