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 and for each , then is Lebesgue measurable.
I am rewriting to verify what you are trying to prove because if , then does not make sense.
Anyway, use subadditivity. . So, given any subset of , you can cover it with a countable union of sets of the form . Then, think about the definition of the Lebesgue measure.