Question: Let $E$ be a set of infinite measure. Let $f$ be a integrable function over $E$, then there necessarily exists a function $g$ which is bounded and measurable such that $f=g$ a.e.
There is a theorem that asserts that a bounded measurable function defined on a set of finite measure is (Lebesgue) integrable. Determining if the above statement is true or false is what I am stuck on.