
Continuity of the translation mapping in L^1
A problem from class. Fix an function
. Define the translation mapping
sending
. How does one prove that
is continuous at
?
We begin by proving it for a dense set. One can either use compactly supported continuous functions, or instead 'very simple functions' (my own terminology). Let us use the latter, but I recommend you to try and prove it using compactly supported continuous function as well. By a 'very simple function', I mean a simple function where the 'very' means that each
is an open interval. Given such a function
, we notice that by the triangle inequality,
However, it's not hard to see that this integral (for a fixed











and therefore
This proves the statement for being a 'very simple function'. Let us now prove it in the general case. Let
be any
function. Fix
and take a 'very simple function'
such that
. By the triangle inequality,
where the last line holds because of the translation invariance of the Lebesgue integral. Now since is a 'very simple function' for which we already proven the translation to be continuous, when taking the limit
we get
This is of course true for all , and therefore
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