concerning fixed point of a measurable function

Question

Let $f:X\longrightarrow X$ be a measurable function and $p\in X$. Let $\delta_p$ denotes the Dirac measure.

If $\delta_p (A)=\delta_{p}(f^{-1}(A))$ for every measurable set $A$, then is $p$ a fixed point of $f$?

Thanks in advance.

Answer 1

The answer is "yes" if and only if for every $x\in X$ there exists a measurable set $B$ that satisfies: $$|B\cap\{p,x\}|=1$$

If $f(p)\neq p$ and $|B\cap\{p,f(p)\}|=1$ then $\delta_p(B)\neq\delta_p(f^{-1}(B))$.

If $f(p)=p$ then it is evident that $\delta_p(A)=\delta_p(f^{-1}(A))$ for every measurable $A$.

---

Remark: if only the Dirac measure is practicized then there is no objection to use $\wp(X)$ as corresponding $\sigma$-algebra. In that case the answer is clearly "yes".