Prove that if f:R->([0,∞]) and f^(-1)((r,∞]) ∈ M for each r∈ M,then f is
Lebesgue measurable.
(M is the σ-algebra of Lebesgue measurable sets)
I think you mean:
Prove that if $\displaystyle f:\mathbb{R} \to [0,\infty]$ and $\displaystyle f^{-1}((r,\infty])\in \mathcal{M}$ for each $\displaystyle r \in \mathbb{R}_{\ge 0}$, then $\displaystyle f$ is Lebesgue measurable.
I am rewriting to verify what you are trying to prove because if $\displaystyle r\in \mathcal{M}$, then $\displaystyle (r,\infty]$ does not make sense.
Anyway, use subadditivity. $\displaystyle (r_1,r_2] = (r_1,\infty]\setminus (r_2,\infty]$. So, given any subset of $\displaystyle [0,\infty]$, you can cover it with a countable union of sets of the form $\displaystyle (r_1,r_2]$. Then, think about the definition of the Lebesgue measure.