Let $m$ be a probability measure on $W \subseteq \mathbb{R}^m$, so that $m(W)=1$.
Consider a measurable function $f:W \rightarrow \mathbb{R}_{\geq 0}$.
Say if the following holds true.
$ \lim_{M \rightarrow \infty} m\left( \{ w \in W : \ f(w) \geq M \} \right) = 0. $
In other words, when $M \rightarrow \infty$, does the set $\{ w \in W : \ f(w) \geq M \}$ necessarily have measure $0$?
If not, provide an example of such $f$; and eventually provide weak conditions on $f$ under which the limit set has measure $0$.