Let f be an extended real valued function defined on a measurable set E of R. Show that f is a measurable function on E iff for each rational number r the set is a measurable subset of E.
-->Suppose f is measurable
then takes opens sets to measurable sets
Fix a rational number r
But = ( , r), which is measurable
<--Suppose for each rational number r the set is a measurable subset of E.
Fix b in
We want to show ( , b) is measurable.
This is clearly true if b is rational (by assumption)
If b is irrational, then we can find a sequence of rationals, { } that are increasing (or decreasing) and converge to b.
Let = E(f < )
Let S = E(f < b)
Clearly, and --> S, and since each is measurable then so is S