# An iteration converges to zero with measure zero

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 $$B_m=\{x\in[0,\frac1m]: f(x), and consider the iteration $$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).