Today’s problem is Problem 6 on the UCLA Fall 2024 Analysis Qual:
Problem 6: Let 
be a measurable set such that 
for all positive integers 
Show that 
or 
where 
is the Lebesgue measure.
Solution: Since the problem requires us to prove a statement with an exception of a measure zero set, we are likely to use measure theoretic machinery. In particular, notice that the above property of 
in fact implies that 
i.e. is 
-invariant. The main tool in this problem turns out to be the Lebesgue differentiation theorem, which says that almost every point of a locally integrable function is a Lebesgue point, i.e. the average value of the function around the point converges to its value at that point.
Now, consider the function 
Pick 
to be a Lebesgue point of 
and note that 
where 
Either way, for any other Lebesgue point 
and family of open balls 
centered at 
one has 
as 
where 
are chosen so that 
and 
shrinks nicely to as 
By the Lebesgue differentiation theorem, we conclude that 
at every Lebesgue point, that is, a.e. If 
this implies and otherwise 
and we are done.
Remark: Note that this also implies that the set of Lebesgue points of is precisely 