I am trying to find what is CGF of this probability measure:
$$\mu=\alpha \delta_a+(1-\alpha) \delta_b$$
I found it difficult because when I tried to calculate Moment generating function, I didn't know what is $\mu(dx)$ (which is density function) but how it looks like :-(.
$$M_X(t)=\int_{\mathbb{R}} \exp{(tx)} \mu(dx)$$