## 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 ) is the measure of the symmetric difference of and . One can easily compute this integral, and show that this symmetric difference is either twice the measure of (whenever is larger than the measure of ) or is 2|h| (when is smaller than the measure of ). Note that this calculations hold because we assumed to be

*an open interval*. Thus, if is small enough (small enough means smaller than the measure of all the 's), we have

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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