I'm trying to prove that $F[x(t)e^{-jat}] = X(w-a)$ using convolution. using the convolution property I know I should get a convolution of $F(x(t))$ and $F(e^{-jat})$ So:
$$ F[x(t)e^{-jat}]= 1/2\pi X(w)*2\pi\delta(w+a) = \int X(\theta)\delta(\omega+a-\theta) d\theta $$ as we know the delta function samples at the impulse moment therefore I get: $$ X(\omega+a) $$ and not $X(\omega-a)$ What am I doing wrong?