Let $\delta_0$ the dirac measure and $\lambda$ the Lebesgue measure.
Suppose we have a probability measure $P$ on $[0,1)$ such that is it invariant under the map $T(x)=nx\;\mbox{mod}\; 1$.
Then $P$ can be written as $$q\delta_0+(1-q)\lambda\quad (*)$$ for $q\in[0,1].$
The proof I am reading say starts by noticing that $\phi_n=\int_{0}^1 e^{i2\pi n x}dP(x)=q$, I am ok with this step.
Next, he argues that $P(\{0\})=q$ and conlude the decomposition $(*)$. Can someone explains why $$P(\{0\})=q\implies (*)\quad?$$