Thread: Measure theory

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.

Originally Posted by Chandru1

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