# Showing that a supremum is a maximum

1. Feb 26, 2012

### whocaresdr

1. The problem statement, all variables and given/known data
Let f be a positive integrable continuous function on ℝ. Fix a measurable set E such that E $\subset$ [0,1]. Let

$s = sup_{β \subset ℝ} [\int_{E} f_{β}(x)dx]$

where

$f_{β}(x) = f(x + β).$

Show that s is actually a maximum (not just a supremum) that is, there is at least one β which gives the supremum.

2. Relevant equations

(I think they're included in the problem statement...)

3. The attempt at a solution

This was an old exam problem from when I took an intro analysis course (Royden) a couple of years ago (2010). I remember I missed something on this exam, but a solution was never posted and the professor never offered one in office hours when I asked. I came across this again and remained puzzled at what exactly had made my proof incorrect... so I'd welcome any thoughts. I know this isn't an attempt per se, but any guidance to start would be great.