# Math Help - Lebesgue measurable

1. ## Lebesgue measurable

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)

2. ## Re: Lebesgue measurable

Originally Posted by Sonifa
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 $f:\mathbb{R} \to [0,\infty]$ and $f^{-1}((r,\infty])\in \mathcal{M}$ for each $r \in \mathbb{R}_{\ge 0}$, then $f$ is Lebesgue measurable.

I am rewriting to verify what you are trying to prove because if $r\in \mathcal{M}$, then $(r,\infty]$ does not make sense.

Anyway, use subadditivity. $(r_1,r_2] = (r_1,\infty]\setminus (r_2,\infty]$. So, given any subset of $[0,\infty]$, you can cover it with a countable union of sets of the form $(r_1,r_2]$. Then, think about the definition of the Lebesgue measure.