Let $\mu$ be a probability measure on compact metric space $X$ and $f:X\to X$ be a homeomorphism and $b=f(a)\neq a$. $\mu$ is called $f$- invariant measure if for every measurable set $A$, $\mu(A)=\mu(f^{-1}(A))$, this implies that if $\mu$ is $f$-invariant and $\mu(A)=1$, then $\mu(f^{-1}(A))=1$, but for Dirac measure $m_b$, we have $m_b(\{b\})=1$ while $m_b(f^{-1}(b))=0$.
Is there a non-atomic measure $\mu$ such that $\mu(A)=1$ but $\mu(f^{-1}(A))\neq 1$? or we can say that for every non-atomic measure $\mu$, we have:
$\mu(A)=1$ implies that $\mu(f^{-1}(A))=1$?
please help me to know it.