Real-Valued Functions

Abstract

Abstract Recall that the measure space given by IR, the Borel subsets of IR, and Lebesgue measure is not a complete measure space. Hence, there exists some Borel set B such that B has zero Lebesgue measure and such that there exists a subset A of B that is not a Borel subset of IR. An example of such a Borel set is the Cantor ternary set C given in Example 1.8. Define a real-valued function f on IR via f(x) = IA(x). Now, consider the sequence {ln}neN of real-valued Borel measurable functions defined on IR via fn(x) = 1/n. Then, off the null set B, fn(x) converges to f(x). Thus, in this situation, fn(x) -+ f(x) almost everywhere, even though I is not Borel measurable. Example 2.4. The composition of a Lebesgue measurable function mapping IR to IR with a continuous function mapping IR to IR need not be Lebesgue measurable.