Due March 2
Let f be integrable on and define, on ,
Then g is integrable on and .
- Let be a closed set such that , and let be the distance from x to F. Then is a Lipschitz function.
- Let Then for any , and for a.e .
There exists and a sequence such that in , but for every x.
Consider the function defined on by
For a fixed enumaeration , let
- F is integrable, and thus the series for F converges for a.e. x.
- F is unbounded on every interval.
- If g = F a.e., then g is unbounded in any interval.