Let be a measurable subset of with and let be a real-valued measurable function on . Prove that for any , there exists a bounded measurable function on such that .
Again, how do I go about doing it? Thanks in advance.
Okay, so what's the main issue here? If you just let , then the measure of the set where they differ is certainly less than . But of course, then may not be bounded as required. So try defining , and looking at what happens to as goes to infinity.