Let $\big(\Bbb R,\mathcal A_{\Bbb R}^*,\overline{\lambda}\big)$ be a lebesgue (complete) measure space and let $f:\Bbb R\to \Bbb R$ be a lebesgue measurable function.
$$\text{I want to guarantee there exists}\;\; M>0\;\; \text{s.t.}\;\; \overline{\lambda}\big(\{x\in\Bbb R:|f(x)|\le M\}\big)>0$$
But the only I've got so far is that $\{x\in\Bbb R:|f(x)|\le M\}=f^{-1}\big([-M,M]\big)\in\mathcal A_{\Bbb R}^*$, since $f$ is measurable.
Any help would be appreciated