Fourier Translation ,How to prove 1 and 2?

Question

I am a beginner in this topics. I am trying to understand Harmonic Analysis. While reading a book got the following lines. Please help me to prove equations 1 and 2.

Let $\mu \in M(R^d)$ and $\tau_y$ denote translation by $y\in R^d$ [$[(\tau_y \mu)(E)=\mu (E-y)]$ for Borel measurable set E]

Then

$\widehat{\tau_y \mu}(\xi)= e^{-2\pi i \xi.y}\widehat{\mu}(\xi)$....1

Let
$e_n(x)=e^{2\pi ix.n}$.

Then $\widehat{e_n \mu}(\xi)=\widehat{\mu}(\xi -n)$..... 2

Answer 1

Observe that for any function $f=f(x)$, $∫_{U} f(x) \, τ_y\mu(dx) = ∫_{U} f(x+y) \, \mu(dx) $. Hence

$$(τ_y \mu)\hat{\phantom{a}}(\xi) = ∫_{ℝ^d} e_\xi(-x) \,τ_y \mu(dx) = ∫_{ℝ^d} e_\xi(-x-y) \, \mu(dx)= ∫_{ℝ^d} e_\xi(-x)e_\xi(-y) \,τ_y \mu(dx) = e_\xi(-y)\hat{\mu}(\xi). $$

The other equation is proven similarly.