If f : X to R(where R is the extended real number system = [infinity,
infinity])
and f^ ((r,infinity]) belongs to script M(m is a measure) for each r
belongs to Q(rational numbers) then f is measurable.

(^ denotes inverse)

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Clarification of Question by madukar-ga on 18 Nov 2002 09:15 PST

this qeustion is in the book real analysis, modern techniques and
their applications, second edition writtern by Gerald B.Folland.page
number : 48,question 4

Hint:

The definition of a measurable function says something about the
inverse image of open sets:  what?  You are given a fact about the
inverse image of (r,+oo].  How can you extend this to a conclusion
about the inverse image of interval (r,s) where both r,s are rational?
 Do the intervals (r,s) for all rational numbers r < s form a basis
for R ?

regards, mathtalk