Let $f(x):[0,+\infty)\to [0,+\infty)$ be a smooth function such that $\mu(f(A))= 0$ if and only if $\mu(A)=0$. ($\mu$ is the Lebesgue measure).

Define $B_m=\{x\in[0,\frac1m]: f(x)<x\}$, and consider the iteration $x_{n+1}=f(x_n)$.

Assuming $\lim\limits_{m\to\infty} m\mu(B_m)=0$, do we have the set $A=\{x_0\in[0,+\infty) : \lim_\limits{n\to\infty} x_n=0\}$ measure zero?

If not, can we add some condition on $f(x)$ to make it true?

Remark: If $\mu(B_m)=0$ for some $m$, the statement is true (click here).


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