Measurable (Extended) Real-Valued Functions For a while, we’ll
mostly be interested in real-valued functions with Lebesgue
measure on the real line, and ultimately in using measure to
give us a new and more general version of integration. When we
couple this with our slightly weakened definition of a
measurable space, this necessitates a slight tweak to our
definition of a measurable function. Given a measurable space
and a function , we define the set as the set of points such
that . We will say that the real-valued function is measurable
if is a measurable subset of for every Borel set of the real
line. We have to treat specially because when we deal with
integration, is special — it’s the additive identity of the real
numbers. The entire real line is a Borel set, and . Thus we find
that must be a measurable subset of . If is another measurable
subset of , then we observe The second term on the right is
either empty or is equal to . And so it’s clear that is
measurable. We say that the function is “measurab