What's the difference of CTFT $\int_{-\infty}^{\infty} f(t) e^{-j \Omega t} dt$ and CTFT of $e^{j \Omega_0 t}$?

Question

What's the difference of CTFT $\int_{-\infty}^{\infty} f(t) e^{-j \Omega t} dt$ and CTFT of $e^{j \Omega_0 t}$?

Because if one sets $f(t)= e^{j \Omega_0 t}$, then shouldn't this be calculated by:

$\int_{-\infty}^{\infty} f(t) e^{-j \Omega t} dt \text{ ?}$

But this gives

$= 2 \pi \delta( \omega - (\Omega_0 - \Omega))$

which is different from $2 \pi \delta(\omega-\Omega_0)=F(\omega)$.

So which one is it?

Or maybe one has to reparametrize so that $\omega_F =\omega+\Omega$?