I have a sense that the countable, dense subset I'm looking for is the step functions, maybe over intervals with rational endpoints, but I'm not sure how to deal with the fact that E is any L-msb set, so there's no guarantee all the intervals are in there.
Analysis professor gave the following problem as a thought exercise:
Show that an Lp function for 1<p<2 can be written as the sum of an L1 and and L2 function.