Fourier transform of product measure

Question

If $\mu_1$ and $\mu_2$ are measures, how does the Fourier transform of $\mu_1 \times \mu_2$ compare to $\hat{\mu}_1$ and $\hat{\mu}_2$?

Answer 1

Hint: Define convolution of measures; try to see how it related to $\mu_{1}\times \mu_{2}$; and it is well-known that, $\widehat{\mu_{1}\ast \mu_{1}}=\hat{\mu_{1}}\hat{\mu_{2}}.$

Answer 2

Let $\phi$ be the Fourier transform of $\mu_1\times \mu_2$. We can compute $\phi$ from the definition of the Fourier transform, using Fubini's theorem: $$ \phi(\xi_1,\xi_2) = \int_{\mathbb R^2} e^{2\pi i x\cdot \xi} \,dx_1\,dx_2 = \int_{\mathbb R} e^{2\pi i x_1 \xi_1} \,dx_1 \cdot \int_{\mathbb R} e^{2\pi i x_2 \xi_2} \,dx_2 = \hat \mu_1(\xi_1)\hat \mu_2(\xi_2) $$ Your convention regarding $2\pi$ in the exponent may vary, but the relation stays the same.